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total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If w(y) is the wife’s income and h(y) is the husband’s income in year y, and we want T to represent the total income, then we can define a new function. T(y) = h(y) + w(y) If thi... |
g(x)_ f (x) g _ ) (x) = (# f g _ ) (x) = (# f g _ ) (x) = (# f g _ ) (x) = x + 1 (# f (x + 1)(x − 1) ____________ x − 1 where x ≠ 1 where x ≠ 1 where x ≠ 1 No, the functions are not the same. g _ ) (x), the condition x ≠ 1 is necessary because when x = 1, the denominator is equal to 0, which makes Note: For (# f the f... |
parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function g takes the input x first and yields an output g(x). Then the function f takes g(x) as an input and yields an output f (g(x)). g(x), the output of g is the input of f (f ° g)(x) = f(g(x)) ... |
(x) = 2x + 1 g(x) = 3 − x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 212 CHAPTER 3 FUNCTIONS Solution Let’s begin by substituting g(x) into f (x). Now we can substitute f (x) into g(x). f (g(x)) = 2(3 − x) + 1 = 6 − 2x + 1 = 7 − 2x g(f (x)) = 3 − (2x + 1) = 3 − 2x − 1 = − 2x + 2 We ... |
expression f (x) takes hours as input and a number of miles driven as the output. The function g(y) requires a number of miles as the input. Using f (x) (miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense. The expression g(f (x)) makes sense, and will yield the numb... |
defines the function g: g(3) = 2. We can then use that result as the input to the function f, so g(3) is replaced by 2 and we get f (2). Then, using the table that defines the function f, we find that f (2) = 8. g(3) = 2 f (g(3)) = f (2) = 8 To evaluate g(f (3)), we first evaluate the inside expression f (3) using the... |
3 2 1 –1 –2 –3 –4 –5 (1, 3) 321 4 5 6 7 x g(1) = 3 Figure 2 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 (3, 6) 321 4 5 6 7 x f (3) = 6 We evaluate g(1) using the graph of g(x), finding the input of 1 on the x-axis and finding the output value of the graph at that input. Here, g(1) = 3. We use this value as the input to the function ... |
using the input value or variable provided. 2. Use the resulting output as the input to the outside function. Example 7 Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input Given f (t) = t2 − t and h(x) = 3x + 2, evaluate f (h(1)). Solution Because the inside expression is h(1), we start ... |
a composite function The domain of a composite function f (g(x)) is the set of those inputs x in the domain of g for which g(x) is in the domain of f. How To… Given a function composition f (g(x)), determine its domain. 1. Find the domain of g. 2. Find the domain of f. 3. Find those inputs x in the domain of g for whi... |
this function is (−∞, 5]. To find the domain of f#∘#g, we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since (−∞, 3] is a proper subset of the domain of f#∘#g. This means the domain of f#∘#g is the same as the domain of g, namely, (−∞, 3]. Analy... |
s (http://openstaxcollege.org/l/compfuncgraph) • Decompose Functions (http://openstaxcollege.org/l/decompfunction) • Composite Function Values (http://openstaxcollege.org/l/compfuncvalue) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 218 CHAPTER 3 FUNCTIONS 3.4 SECTION EXERCISES VERBAL ... |
Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.4 SECTION EXERCISES 219 22. fx= gx= − x x +x a. g"∘"f x b. g"∘"f 23. px = " mx=x − — x √ a. px mx b. pmx c. mpx 24. qx= hx=x − — x √ x gx=√ 25. fx= — x− f"∘"gx a. qx hx b. qhx c. hqx, fif xgxhx=f x 26. hx=x+ 27. hx=x− 30. hx=+ √ —... |
− − Table 4 − − − 66. f"∘"g 69. g"∘"f 67. f"∘"g 70. g"∘"g 68. g"∘"f 71. f"∘"f o fif ggf 72. fx=x+gx=−x 73. fx=x+gx=−x 74. fx=√ — x+gx=−x 75. fx= gx=x+ _ x +" f x=x+gx=x +fi 76. fg 77. fgx 78. g(f− 79. g"∘"g x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.4 SECTION EXERCISES 2... |
. NTt b. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 222 CHAPTER 3 FUNCTIONS LEARNING OBJECTIVES In this section, you will: • ecreate the library of functions. • Graph and describe functions using ertical and horiontalshifts. • Graph and describe functions using reflections about the ... |
3 4 5 21 x –5 –4 –3 –2 5 4 3 2 1 –1–1 –2 –3 –4 –5 Figure 2 Vertical shift by k = 1 of the cube root function f ( x) = 3 √ — x. To help you visualize the concept of a vertical shi ft, consider that y = f (x). Therefore, f (x) + k is equivalent to y + k. Every unit of y is replaced by y + k, so the y-value increases or ... |
that, with a vertical shift, the input values stay the same and only the output values change. See Table 1. t V(t) S(t) 0 0 20 8 0 20 17 220 240 19 0 20 24 0 20 10 220 240 Table 1 How To… Given a tabular function, create a new row to represent a vertical shift. 1. Identify the output row or column. 2. Determine the ma... |
in what is known as a horizontal shift, shown in Figure 5. f(x) –5 –4 –3 –2 5 4 3 2 1 –1–1 –2 –3 –4 –5 f(x+1) f(x) 21 3 4 5 x Figure 5 Horizontal shift of the function f (x) = 3 √ — x. Note that h = +1 shifts the graph to the left, that is, towards negative values of x. For example, if f (x) = x2, then g(x) = (x − 2)2... |
(t + 2). V 300 250 240 200 150 100 50 –2 0 50– – –4 Left 2 4 8 12 16 20 24 28 t Figure 6 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 226 CHAPTER 3 FUNCTIONS Analysis Note that V(t + 2) has the effect of shifting the graph to the left. Horizontal changes or “inside changes” affect the ... |
4 shifted to 7, 6 shifted to 9, and 8 shifted to 11. Analysis Figure 7 represents both of the functions. We can see the horizontal shift in each point. y 12 10 8 6 4 2 –2–2 –4 –6 –8 –10 –12 –12 –10 –8 –6 –4 42 6 8 10 12 x f (x) g (x) = f(x – 3) Figure 7 Download the OpenStax text for free at http://cnx.org/content/col... |
our input values. Example 6 Interpreting Horizontal versus Vertical Shifts The function G (m) gives the number of gallons of gas required to drive m miles. Interpret G (m) + 10 and G (m + 10). Solution G (m) + 10 can be interpreted as adding 10 to the output, gallons. This is the gas required to drive m miles, plus an... |
a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in f (x + 1) − 3 is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9. Let us follow one point of the graph of f (x) =. x.. • The po... |
can be applied to a function is a reflection over the x- or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. The reflections are shown in Figure 12. y Horizontal reflection f(x) Original function f(–x) x Vertical... |
V(t) = − √ — t Notice that this is an outside change, or vertical shift, that affects the output s(t) values, so the negative sign belongs outside of the function. b. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14. s(t) 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 ... |
opposite of the original input value and the h(x) values stay the same as the f (x) values. See Table 8. x h(x) –2 1 –4 3 Table 8 –6 7 –8 11 Try It #6 A function f (x) is given as Table 9. Create a table for the functions below. x f(x) −2 5 0 10 Table 9 2 15 4 20 a. g(x) = −f (x) b. h(x) = f (−x) Download the OpenStax... |
–3 –4 –5 (a) 1 2 3 4 5 t –5 –4 –3 –2 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 t –5 –4 –3 –2 (b) Figure 16 k(t) 5 4 3 2 1 –1 –1 –2 –3 –4 –5 (c) 21 3 4 5 t Analysis As a model for learning, this function would be limited to a domain of t ≥ 0, with corresponding range [0, 1). Try It #7 Given the toolkit function f (x) = x 2,... |
) (c) (c) Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function. We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd fun... |
in Figure 18. Notice that the graph is symmetric about the origin. For every point (x, y) on the graph, the corresponding point (−x, −y) is also on the graph. For example, (1, 3) is on the graph of f, and the corresponding point (−1, −3) is also on the graph. f (x) f 5 4 3 2 1 –1 –2 –3 –4 –5 (−1, −3) –1 –2 –3 –4 –5 (1... |
the value of a. 2. Multiply all range values by a. 3. If a > 1, the graph is stretched by a factor of a. If 0 < a < 1, the graph is compressed by a factor of a. If a < 0, the graph is either stretched or compressed and also reflected about the x-axis. Example 13 Graphing a Vertical Stretch A function P(t) models the p... |
for the function g(x) = #1 __ f (x). 2 x f (x) 2 1 6 7 8 11 4 3 Table 10 Solution The formula g(x) =# #1 __ f (x) tells us that the output values of g are half of the output values of f with the same 2 inputs. For example, we know that f (4) = 3. Then We do the same for the other values to produce Table 11. (3) =# #3 ... |
4 Try It #10 Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units. Horizontal Stretches and Compressions Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a ... |
in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population. Solution Symbolically, we could write R(1) = P(2), R(2) = P(4), and in general, R(t) = P(2t). See Figure 24 for a graphical compa... |
4 –2 –4 –6 –8 –10 –12 (b) 84 12 16 20 x Analysis Because each input value has been doubled, the result is that the function g(x) has been stretched horizontally by a factor of 2. Example 18 Recognizing a Horizontal Compression on a Graph Relate the function g(x) to f (x) in Figure 26. f (x) 6 5 4 3 2 1 – –1 1 Figure 26... |
a little trickier to think about. When we write g(x) = f (2x + 3), for example, we have to think about how the inputs to the function g relate to the inputs to the function f. Suppose we know f (7) = 12. What input to g would produce that output? In other words, what value of x will allow g(x) = f (2x + 3) = 12? We wo... |
3.5 TRANSFORMATION OF FUNCTIONS 241 Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See Table 17. x 2f (3x) 2 20 4 28 Table 17 6 30 8 34 Finally, we can apply the vertical shift, which will add 1 ... |
staxcollege.org/l/functrans) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.5 SECTION EXERCISES 243 3.5 SECTION EXERCISES VERBAL 1. 2. x y 3. When examining the formula of a function that 4. When examining the formula of a function that is is the result of multiple transformati... |
– – – – 40. y f – – – – –– – – – – x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 3.5 SECTION EXERCISES 245 41. 44. y f – – – – –– – – – – 42. x y f x – – – – –– – – – – 43. y f – – – – –– – – – – x y – – – – –– – – – – f x 45. fx=x 48. fx=x− 46. gx=√x — 49. gx=x 47. hx=__ x +... |
759/latest. SECTION 3.6 ABSOLUTE VALUE FUNCTIONS 247 LEARNING OBJECTIVES In this section you will: • Graph an absolute value function. • Solve an absolute value equation. 3.6 ABSOLUTE VALUE FUNCTIONS Figure 1 Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms ... |
so, with the resistance R in ohms, | R − 680 | ≤ 34 Try It #1 Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 248 CHAPTER 3 FUNCTIONS Graphing an Absolute Value Fun... |
5 4 3 2 1 –5 –4 –3 –2 –1–1 –2 –3 –4 –5 Ratio 1/1 x 21 3 4 5 (3, −2) From this information we can write the equation Figure 6 f (x) = 2| x − 3 |#− 2, treating the stretch as a vertical stretch, or f (x) = | 2(x − 3) |#− 2, treating the stretch as a horizontal compression. Analysis Note that these equations are algebrai... |
absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points. Solving an Absolute Value Equation In Other Types of Equations, we touched on the concepts of absolu... |
4x +#1 6 = 4x −8 =#4x −8 ___ 4 6 __ x = = 1.5 4 The function outputs 0 when x = 1.5 or x = −2. See Figure 8. = −2 x = Substitute 0 for f (x). Isolate the absolute value on one side of the equation. Break into two separate equations and solve. y 10 8 6 4 2 –0.5 –2 –4 –6 –8 –10 (−2, 0) –1 –1.5 –2 –2.5 x where f(x) = 0 f ... |
which the distance that point x is from 10 is at least 15 units. Express this using absolute value notation. 8. Find all function values f (x) such that the distance from f (x) to the value 8 is less than 0.03 units. Express this using absolute value notation. For the following exercises, find the x- and y-intercepts ... |
graphing utility to graph f (x) = 10| x − 2| on the viewing window [0, 4]. Identify the corresponding range. Show the graph. 33. Use a graphing utility to graph f (x) = −100| x|#+ 100 on the viewing window [−5, 5]. Identify the corresponding range. Show the graph. For the following exercises, graph each function using... |
function is one-to-one. • erify the inerse using composition. • Find the inerse of a function. • etermine the domain and range of an inerse function and restrict the domain of a function to mae it one-to-one. • Use the graph of a one-to-one function to graph its inerse function on the same aes. • Use a graph or table ... |
the equation for F after substituting a value for C. For example, to convert 26 degrees Celsius, she could write 26 =# #5 __ (F − 32) 9 9 __ = F − 32 26 ċ 5 9 __ + 32 ≈ 79 F = 26 ċ 5 After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully... |
1(4x) =# #1 __ (4x) = x 4 1 1 __ __ (f ∘ f −1)(x) = f (# x ) = x x ) = 4 (# 4 4 A few coordinate pairs from the graph of the function y = 4x are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs 1 __ x are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each from the graph of the function y =... |
(x, f (x)) (2, 4) (5, 12) (x, g (x)) (4, 2) (12, 5) Table 1 Try It #1 Given that h−1(6) = 2, what are the corresponding input and output values of the original function h? How To… Given two functions f (x) and g (x), test whether the functions are inverses of each other. 1. Determine whether f (g(x)) = x or g(f (x)) =... |
? Finding Domain and Range of Inverse Functions The outputs of the function f are the inputs to f −1, so the range of f is also the domain of f −1. Likewise, because the inputs to f are the outputs of f −1, the domain of f is the range of f −1. We can visualize the situation as in Figure 3. Domain of f Range of f f (x)... |
). If f (x) = (x − 1)2 on [1, ∞), then the inverse function is f −1(x) = √ x + 1. — • The domain of f = range of f −1 = [1, ∞). • The domain of f −1 = range of f = [0, ∞). Q & A… Is it possible for a function to have more than one inverse? No. If two supposedly different functions, say, g and h, both meet the definitio... |
x f (x) = √ — x f (x) =.x. Table 2 Solution The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no meaningful inverse. The absolute value function can be restricted to the domain [0, ∞), where it is equal to the identity ... |
70). Interpreting the Inverse of a Tabular Function t (minutes) f (t) (miles) 30 20 50 40 70 60 90 70 Table 3 Solution The inverse function takes an output of f and returns an input for f. So in the expression f −1(70), 70 is an output value of the original function, representing 70 miles. The inverse will return the c... |
g(x) is given in Figure 5. Find g(3) and g−1(3). g(x) 6 5 4 3 2 1 – –1 1– 1 2 3 4 5 6 7 x Figure 5 Solution To evaluate g(3), we find 3 on the x-axis and find the corresponding output value on the y-axis. The point (3, 1) tells us that g(3) = 1. To evaluate g −1(3), recall that by definition g −1(3) means the value of... |
to represent the conversion because the input and output variables are descriptive, and writing C−1 could get confusing. Try It #7 1 __ (x − 5) Solve for x in terms of y given y = 3 Example 8 Solving to Find an Inverse Function Find the inverse of the function f (x) = 2 ____ x − 3 + 4. Solution y = + 4 Set up an equat... |
we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function f (x) = x2 restricted to the domain [0, ∞), on which this function is one-to-one, and graph it as in Figure 7. f (x) 5 4 3 2 1 –1–1 –2 –5 –4 –3 –2 21 3 4 5 x Figure 7 Quadratic fun... |
), so the inverse will have a domain of (−∞, ∞) and range of (0, ∞). If we reflect this graph over the line y = x, the point (1, 0) reflects to (0, 1) and the point (4, 2) reflects to (2, 4). Sketching the inverse on the same axes as the original graph gives Figure 10. y 5 4 3 2 1 –1–1 –2 –3 –4 –5 –5 –4 –3 –2 f –1(x) y... |
. fx= x_____+ x+ 18. 22. 27. + fx=√x — − fx=+ √ x fx= x_____− x+ 19. fx=√−x — 23. 28. — fx=−√ x x______+ −x fx= 20. 24. 29. — fx= fx=+ √x− _____ x+ x______+ fx= −x 30. fx=x + x−∞ 31. fx=x + x+ −∞ 32. fx=x −x+ ∞ fif. Thn fif 33. fx=x+ 34. fx=x− 35. fx=x − 36. fx= x − gx= x − x a. fgxg fx b. fxgx f xgx 37. fx=√ x− g x=x ... |
. fx=x − 69. _____ x− 70. fx= _ x− REAL-WORLD APPLICATIONS 71. xy __ x+ fx= 72. ThC Cr=πr rCrπ 73. Th tdt=t td t Download the OpenStax text for free at http://cnx.org/content/col11759/latest. CHAPTER 3 REVIEW 267 CHAPTER 3 REVIEW Key Terms absolute maximum the greatest value of a function over an interval absolute mini... |
f (x), the inverse is a function f −1(x) such that f −1(f (x)) = x for all x in the domain of f; this also implies that f ( f −1(x)) = x for all x in the domain of f −1 local extrema collectively, all of a function’s local maxima and minima local maximum a value of the input where a function changes from increasing to... |
2 f (x) = x3 1_ f (x) = x 1 _ x2 f (x) = √ — x f (x) = ∆y _ ∆x = — x 3 √ f (x2) − f (x1) _ x2 − x1 Composite function (f ∘#g)(x) = f (g(x)) Vertical shift g(x) = f (x) + k (up for k > 0) Horizontal shift g(x) = f (x − h) (right for h > 0) Vertical reflection g(x) = −f (x) Horizontal reflection g(x) = f (−x) Vertical st... |
value. See Example 13. • A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See Example 14. • The graph of a one-to-one function passes the horizontal line test. See Example 15. 3.2 Domain and Range • The domain of a function includes all real input va... |
http://cnx.org/content/col11759/latest. 270 CHAPTER 3 FUNCTIONS • A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values. • Minima and maxima are also called ext... |
input by –1. • A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See Example 9. • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Exampl... |
of its domain. • For a tabular function, exchange the input and output rows to obtain the inverse. See Example 5. • The inverse of a function can be determined at specific points on its graph. See Example 6. • To find the inverse of a formula, solve the equation y = f (x) for x as a function of y. Then exchange the la... |
5 x For the following exercises, graph the functions. 12. f (x) =.x + 1. 13. f (x) = x 2 − 2 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. CHAPTER 3 REVIEW 273 For the following exercises, use Figure 2 to approximate the values. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 14. f (2) 15. f... |
30. Find the local minimum of the function graphed in Exercise 27. 31. Find the local extrema for the function graphed in Exercise 28. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 274 CHAPTER 3 FUNCTIONS 32. For the graph in Figure 3, the domain of the function is [−3, 3]. The range i... |
. f (x) = 4[.x − 2. − 6] 52. f (x) = −(x + 2)2 − 1 47. f (x) = √ — x + 5 50. f (x) = 5 √ — −x − 4 For the following exercises, sketch the graph of the function g if the graph of the function f is shown in Figure 4. y 5 4 3 2 1 –1 –1 –2 –5 –4 –3 –2 21 3 4 5 x Figure 4 53. g(x) = f (x − 1) 54. g(x) = 3f (x) Download the ... |
10 8 6 4 2 –2 –2 –4 –6 –8 –10 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 42 6 8 10 x 21 3 4 5 x For the following exercises, graph the absolute value function. 66. f (x) = | x − 5 | 67. f (x) = −| x − 3 | 68. f (x) = | 2x − 4 | Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 276 CHAPTER 3 FUNCTIONS I... |
Write the domain of the function f (x) = √ — 3 − x in interval notation. 7. Given f (x) = 2x 2 − 5x, find f (a + 1) − f (1). 8. Graph the function f (x) = { x + 1 if −2 < x < 3 x ≥ 3 −x if 9. Find the average rate of change of the function f (x) = 3 − 2x 2 + x by finding f (b) − f (a) _ b − a. For the following exerci... |
org/content/col11759/latest. 278 CHAPTER 3 FUNCTIONS For the following exercises, use the graph of the piecewise function shown in Figure 2. 25. Find f (2). 26. Find f (−2). 27. Write an equation for the piecewise function. y 5 4 3 2 1 f 21 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 Figure 2 For the following exercises, use... |
/latest. 280 CHAPTER 4 LINEAR FUNCTIONS LEARNING OBJECTIVES In this section, you will: • Represent a linear function. • Determine whether a linear function is increasing, decreasing, or constant. • Interpret slope as a rate of change. • Write and interpret an equation for a linear function. • Graph linear functions. • ... |
. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station. 7 http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm Download the OpenStax text for free at http://cnx... |
ant, or slope of the line. The point at which the input value is zero is the vertical intercept, or y-intercept, of the line. We can see from the graph that the y-intercept in the train example we just saw is (0, 250) and represents the distance of the train from the station when it began moving at a constant speed 500... |
This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases. Determining Whether a Linear Function Is Increasing, Decreasing, or Constant The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be in... |
unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month. Solution Analyze each function. a. The function can be represented as f (x) = 60x where x is the number of days. The slope, 60, is positive so the function... |
OpenStax text for free at http://cnx.org/content/col11759/latest. 284 1. CHAPTER 4 LINEAR FUNCTIONS x2 – x1 y2 – y1 y 10 x2, y2) (x1, y1) 0 –1 321 4 ∆y ∆x = y2 – y1 x2 – x1 m = x Figure 6 The slope of a function is calculated by the change in y divided by the change in x. It does not matter which coordinate is used as... |
slope of the line as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used. Note that if we had reversed them, we would have obtained the same slope. m = (−2) − (1) _________ 3 − 8 = −3 ___ −5 3 _ = 5 Download the OpenStax text for free at http://cnx.org/co... |
two points on the line to find the slope. Let’s choose (0, 7) and (4, 4). We can use these points to calculate the slope. m = y2 − y1 _ x2 − x1 = 4 − 7 _____ 4 − 0 = − #3 _ 4 Now we can substitute the slope and the coordinates of one of the points into the point-slope form. y − y1 = m(x − x1) y − 4 = − #3 _ (x − 4) 4 ... |
− y1 _ x2 − x1 −4 − 2 _______ −2 − 0 −6 ___ −2 = = Substitute the slope and the coordinates of one of the points into the point-slope form. = 3 y − y1 = m(x − x1) y − (−4) = 3(x − (−2)) y + 4 = 3(x + 2) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 4.1 LINEAR FUNCTIONS 287 We c... |
the points to calculate the slope. f (3) = −2 → (3, −2) f (8) = 1 → (8, 1) m = y2 − y1 _ x2 − x1 1 − (−2) ________ 8 − 3 = Substitute the slope and the coordinates of one of the points into the point-slope form. = #3 _ 5 y − y1 = m(x − x1) y − (−2) =# #3 _ (x − 3) 5 Download the OpenStax text for free at http://cnx.or... |
by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that m = 15. We can substitute the initial value and the rate of change into the slope-intercept form of a line. f (x) = mx + b ↑ ↑ 15 200 N(t) = 15t + 200 Figure 10 We can write the formula N(t) = 15t + 200. With this formula, we ca... |
. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold. We can now write the final equation. I(n) = 80n + 520 Our final interpretation is that Ilya’s base salary is $520 per week and he earns an additional $80 commission for each policy sold. Example 10 Using ... |
.5 13.5 14.5 16.5 18.5 Table 2 Graphing Linear Functions Now that we’ve seen and interpreted graphs of linear functions, let’s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by... |
5 = 1 ⇒ (6, 1) 3 Plot the coordinate pairs and draw a line through the points. Figure 11 represents the graph of the function 2 __ x + 5. f (x) = − 3 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 4.1 LINEAR FUNCTIONS 291 f(x) (0, 5) f 6 5 4 3 2 1 (3, 3) (6, 1) – – __ x + 5. Fig... |
↑Run = 2 1 2 4 3 Figure 12 x 5 6 7 graphical interpretation of a linear function In the equation f (x) = mx + b • b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis. • m is the slope of the line and indicates the vertical displacement (rise) and horizontal displaceme... |
the left and right by repeating, and then draw a line through the points. f f(x) 6 5 4 3 2 1 – – __ x + 5 and shows how to calculate the rise over run for the slope. Figure 13 Graph of f (x ) = − 3 Analysis The graph slants downward from left to right, which means it has a negative slope as expected. Try It #6 Find a ... |
illustrates vertical shifts of the function f (x) = x. Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method. H... |
of a Line Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 18. We can see right away that the graph crosses the y-axis at the point (0, 4) so this is the y-intercep... |
different y-intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through (0, 3) so f must be represented by line I. b. This function also has a slope of 2, but a y-intercept of −3. It must pass through the point (0, −3) and slant upward from left to right. It must be ... |
). Q & A… Do all linear functions have x-intercepts? No. However, linear functions of the form y = c, where c is a nonzero real number, are the only examples of linear functions with no x-intercept. For example, y = 5 is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in Figure 2... |
21 3 4 5 x Figure 23 A horizontal line representing the function f (x) = 2. A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not r... |
28 are parallel lines: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the y-intercept of the other, they would become coincident1 –1 –2 –3 –6 –5 –4 –3 –2 21 3... |
) = −4x + 3 negative reciprocal of −4 is 4 1 −4 ( ) = −1 _ 4 parallel and perpendicular lines Two lines are parallel lines if they do not intersect. The slopes of the lines are the same. f (x) = m1x + b1 and g(x) = m2x + b2 are parallel if m1 = m2. If and only if b1 = b2 and m1 = m2, we say the lines coincide. Coincide... |
− 4 and Writing the Equation of a Line Parallel or Perpendicular to a Given Line If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line. Writing Equations of Parallel Lines Suppose for example, we are given the e... |
cept form, we can substitute m = 3, x = 3, and f (x) = 0 into the slope-intercept form to find the y-intercept. g(x) = 3x + b 0 = 3(3) + b b = −9 The line parallel to f (x) that passes through (3, 0) is g(x) = 3x − 9. Analysis We can confirm that the two lines are parallel by graphing them. Figure 31 shows that the two... |
__ x + 2 is perpendicular to f (x) = 2x + 4 and passes through the point (4, 0). Be aware that perpendicular 2 lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature. Q & A… A horizontal line has a slope of zero and a vertical line has an undefined slope. These two li... |
h(x) = 2x − 4, write an equation for the line passing through (0, 0) that is a. parallel to h(x) b. perpendicular to h(x) How To… Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point. 1. Determine the slope of the line passing through the points. 2. F... |
org/content/col11759/latest. 30 4 CHAPTER 4 LINEAR FUNCTIONS 4.1 SECTION EXERCISES VERBAL 1. Terry is skiing down a steep hill. Terry’s elevation, E(t), in feet after t seconds is given by E(t) = 3000 − 70t. Write a complete sentence describing Terry’s starting elevation and how it is changing over time. 2. Jessica is ... |
two given points. 24. (2, 4) and (4, 10) 27. (8, −2) and (4, 6) 25. (1, 5) and (4, 11) 28. (6, 11) and (−4, 3) 26. (−1, 4) and (5, 2) For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. 29. f (−5) = −4, and f (5) = 2 30. f (−1) = 4 and f (5) = 1 31... |
71) 48. Line 1: Passes through (−8, −55) and (10, 89) Line 2: Passes through (9, −44) and (4, −14) 49. Line 1: Passes through (2, 3) and (4, −1) Line 2: Passes through (6, 3) and (8, 5) 50. Line 1: Passes through (1, 7) and (5, 5) Line 2: Passes through (−1, −3) and (1, 1) 51. Line 1: Passes through (2, 5) and (5, −1) ... |
OpenStax text for free at http://cnx.org/content/col11759/latest. 30 6 61. CHAPTER 4 LINEAR FUNCTIONS y 6 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2 62. 321 4 5 x –6 –5 –4 –3 –2 y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 63. 321 4 5 6 x –6 –5 –4 –3 –2 y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 321 4 5 6 x For the following exercises, mat... |
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