4%2d7 inverse linear functions c4 l7...write the final equation in slope -lqwhufhswirup so, ....

Find the inverse of each relation.��{(4, ±15), (±8, ±18), (±2, ±16.5), (3, ±15.25)}

62/87,21��To find the inverse, exchange the coordinates of the ordered pairs.� (4, ±��ĺ��±�� (±8, ±��ĺ��±18, ±8) � (±2, ±��ĺ��±16.5, ±2) � (3, ±��ĺ��±15.25, 3) � The inverse is {(í15, 4), (í18, í8), (í16.5, í2), (í15.25, 3)}.

��

62/87,21��Write the coordinates as ordered pairs. Then exchange the coordinates.� (±��ĺ��±3) � ��ĺ�� ĺ�� (6, ±��ĺ��±12.5, 6) � The inverse is {(11.8, í3), (3.7, 0), (1, 1), (í12.5, 6)}.

Graph the inverse of each relation.

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4-7 Inverse Linear Functions


��

62/87,21��The graph of the relation passes through the points at (-5, 1), (0, 2), and (5, 3). � To find points through which the inverse passes, exchange the coordinates of the ordered pairs� (±��ĺ��±�� ĺ�� ĺ�� Graph these points and then draw a line that passes through them. �

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62/87,21��The graph of the relation passes through the points at (±2, 1) and (2, ±2). � To find points through which the inverse passes, exchange the coordinates of the ordered pairs.� (±��ĺ��±2) � (2, ±��ĺ��±2, 2) � Graph these points and then draw a line that passes through them. �

Find the inverse of each function.



Find the inverse of each function.��f (x) = ±2x + 7

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� Write the final equation in slope-LQWHUFHSW�IRUP��

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Write the final equation in slope-intercept form. So, .

��CCSS REASONING Dwayne and his brother purchase season tickets to the Cleveland Crusaders games. The ticket package requires a one-time purchase of a personal seat license costing $1200 for two seats. A ticket to each game costs $70. The cost C(x) in dollars for Dwayne for the first season is C(x) = 600 + 70x, where x is the numberof games Dwayne attends. � a. Find the inverse function. � b. What do x and C±1(x) represent in the context of the inverse function? � c. How many games did Dwayne attend if his total cost for the season was $950?

62/87,21��a.

� b. x is Dwayne¶s total cost, and C -1(x) is the number of games Dwayne attended.�

-1 eSolutions Manual - Powered by Cognero


� c. Evaluate C -1(950). �

� So, Dwayne attended 5 games.

Find the inverse of each relation.��

62/87,21��To find the inverse, exchange the coordinates of the ordered pairs.� (±��ĺ��±5) � ��ĺ�� ĺ�� (±��ĺ��±10) � The inverse is {(13, ±5), (10.8, 6), (11.4, 3), (14, ±10)}.

��{(±4, ±49), (8, 35), (±1, ±28), (4, 7)}

62/87,21��To find the inverse, exchange the coordinates of the ordered pairs.� (±4, ±��ĺ��±49, ±4) � ��ĺ�� (±1, ±��ĺ��±28, ±1) � ��ĺ�� The inverse is {(±49, ±4), (35, 8), (±28, ±1), (7, 4)}.



��

62/87,21��Write the coordinates as ordered pairs. Then exchange the coordinates of each pair.� (±8, ±��ĺ��±36.4, ±8) � (±2, ±��ĺ��±15.4, ±2) � (1, ±��ĺ��±4.9, 1) � ��ĺ�� ĺ�� The inverse is {(±36.4, ±8), (±15.4, ±2), (±4.9, 1), (9.1, 5), (30.1, 11)}.

��

62/87,21��Write the coordinates as ordered pairs. Then exchange the coordinates of each pair.� (±��ĺ��±3) � (±��ĺ��±�� ĺ�� (3, ±��ĺ��±2.8, 3) � (5, ±��ĺ��±6.2, 5) � The inverse is {(7.4, ±3), (4, ±1), (0.6, 1), (±2.8, 3), (±6.2, 5)}.





��

62/87,21��The graph of the relation passes through the points at (±3, ±2), (0, ±1), and (3, 0). � To find points through which the inverse passes, exchange the coordinates of the ordered pairs� (±3, ±��ĺ��±2, ±3) � (0, ±��ĺ��±�� ĺ�� Graph these points and then draw a line that passes through them. �



��

62/87,21��The graph of the relation passes through the points at (±4, ±3), (±2, ±2), (0, ±1), (2, 0), and (4, 1)� To find points through which the inverse passes, exchange the coordinates of the ordered pairs. � (±4, ±��ĺ��±3, ±4) � (±2, ±��ĺ��±2, ±2) � (0, ±��ĺ��±1, 0) � ��ĺ�� ĺ�� Graph these points and then draw a line that passes through them. �

Find the inverse of each function.



Find the inverse of each function.��f (x) = 25 + 4x

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So, .

��

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� So, .

��f (x) = 4(x + 17)



��f (x) = 4(x + 17)

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So, .

��f (x) = 12 ± 6x



��f (x) = 12 ± 6x

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So, .

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So, .

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��DOWNLOADS An online music subscription service allows members to download songs for $0.99 each after



��DOWNLOADS An online music subscription service allows members to download songs for $0.99 each after paying a monthly service charge of $3.99. The total monthly cost C(x) of the service in dollars is C(x) = 3.99 + 0.99x, where x is the number of songs downloaded. � a.�)LQG�WKH�LQYHUVH�IXQFWLRQ�� b. What do x and C±1(x) represent in the context of the inverse function? c. How many songs were downloaded if a member¶s monthly bill is $27.75?

62/87,21��a.

� � b. x is the total monthly cost of the service, and C -1(x) is the number of songs downloaded.� c. Evaluate C -1(27.75). �

� So, 24 songs were downloaded.

��LANDSCAPING At the start of the mowing season, Chuck collects a one-time maintenance fee of $10 from his



��LANDSCAPING At the start of the mowing season, Chuck collects a one-time maintenance fee of $10 from his customers. He charges the Fosters $35 for each cut. The total amount collected from the Fosters in dollars for the season is C±1(x) = 10 + 35x, where x is the number of times Chuck mows the Fosters¶�lawn. � a.�)LQG�WKH�LQYHUVH�IXQFWLRQ�� b. What do x and C±1(x) represent in the context of the inverse function? c. How many times did Chuck mow the Fosters¶�lawn if he collected a total of $780 from them?

62/87,21��a.

� b. is the total amount collected from the Fosters, and C -1(x) is the number of times Chuck mowed the Fosters¶�lawn. � c. Evaluate C -1(780). �

� So, Chuck mowed the Foster¶s lawn 22 times.

Write the inverse of each equation in f ±1(x) notation.



Write the inverse of each equation in f ±1(x) notation.��3y ± 12x = ±72

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��x + 5y = 15

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��±42 + 6y = x

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��3y + 24 = 2x

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��3y ± x = 3

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CCSS TOOLS Match each function with the graph of its inverse.



CCSS TOOLS Match each function with the graph of its inverse.

��f (x) = x + 4

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� This equation is shown on graph D.

��f (x) = 4x + 4



��f (x) = 4x + 4

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� This equation is shown on graph B.�

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� This equation is shown on graph C.

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� This equation is shown on graph A.�

Write an equation for the inverse function f -1(x) that satisfies the given conditions.eSolutions Manual - Powered by Cognero


Write an equation for the inverse function f -1(x) that satisfies the given conditions.��slope of f (x) is 7; graph of f ±1(x) contains the point (13, 1)

62/87,21��Write an equation for f (x) in terms of x and y . �

� Find the inverse of f (x). �

��graph of f (x) contains the points (í3, 6) and (6, 12)



��graph of f (x) contains the points (í3, 6) and (6, 12)

62/87,21��

If the graph of f (x) contains the points (í3, 6) and (6, 12), then the graph of f í1 (x) contains the points (6, í3) and (12, 6). Find the slope of the line that passes through these points. �

� Choose (12, 6) and find the y-intercept of the line. �

The line that passes through (6, í3) and (12, 6) is �$Q�HTXDWLRQ�IRU�f -1 (x) is .

��graph of f (x) contains the point (10, 16); graph of f ±1(x) contains the point (3, ±16)

62/87,21��

If the graph of f (x) contains the point (10, 16), then the graph of f í1 (x) contains the point (16, 10). Find the slope of the line that passes through (16, 10) and (3, ±16). �

� Choose (3, ±16) and find the y-intercept of the line. �

� The line that passes through (16, 10) and (3, ±16) is �$Q�HTXDWLRQ�IRU�f -1 (x) is .

±1



��slope of f (x) is 4; f ±1(5) = 2

62/87,21��Write an equation for f (x) in terms of x and y . �

� Find the inverse of f (x). �

�

��CELL PHONES Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets



��CELL PHONES Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets billed an additional charge for every minute she uses the phone past the 700 minutes. During her first month, Mary Ann used 26 additional minutes and her bill was $37.79. During her second month, Mary Ann used 38 additional minutes and her bill was $41.39. � a. Write a function that represents the total monthly cost C(x) of Mary Ann¶s cell phone package, where x is the number of additional minutes used. � b.�)LQG�WKH�LQYHUVH�IXQFWLRQ�� c. What do x and C±1(x) represent in the context of the inverse function? � d. How many additional minutes did Mary Ann use if her bill for her third month was $48.89?

62/87,21��a. C(x) = 0.3x + 29.99 � b.

� c. x is Mary Ann¶s total monthly cost, and C-1(x) is the number of additional minutes used.� d. Evaluate C-1(48.89). �

� So, Mary Ann used 63 additional minutes.

��MULTIPLE REPRESENTATIONS�,Q�WKLV�SUREOHP��\RX�ZLOO�H[SORUH�WKH�GRPDLQ�DQG�UDQJH�RI�LQYHUVH�IXQFWLRQV�� a. Algebraic Write a function for the area A(x��RI�WKH�UHFWDQJOH�VKRZQ�� b. Graphical Graph A(x). Describe the domain and range of A(x) in the context of the situation. c. Algebraic Write the inverse of A(x). What do x and A±1(x) represent in the context of the situation? eSolutions Manual - Powered by Cognero


d. Graphical Graph A±1(x). Describe the domain and range of A±1(x) in the context of the situation. e. Logical Determine the relationship between the domains and ranges of A(x) and A±1(x).

62/87,21��a. A(x) = 8(x í�3) or A(x) = 8x í�24 � b.

� Sample answer: The domain represents possible values of x. The range represents the area of the rectangle and must be positive. This means that the domain of A(x) is all real numbers greater than 3, and the range of A(x) is all positive real numbers.

c. ; x is the area of the rectangle and A-1(x) is the value of x in the expression for the length of the

VLGH�RI�WKH� rectangle x í�3. � d.

� Sample answer: The domain represents the area of the rectangle and must be positive. The range represents possiblevalues for x in the expression x - 3. This means that the domain of A-1(x ) is all positive real numbers, and the range of A-1(x) is all real numbers greater than 3. � e . Sample answer: The domain of A(x) is the range of Aí1(x), and the range of A(x) is the domain of Aí1(x).

CHALLENGE ±1



��CHALLENGE If f (x) = 5x + a and f ±1 (10) = ±1, find a.

62/87,21��Find the inverse of f (x). �

� If f -1(10) = ±1, then an ordered pair of f -1(x) is (10, ±1). Evaluate f -1(x) for (10, ±1).�

��CHALLENGE If and , find a and b.

62/87,21��

Write the inverse of .

�

� So, . Compare this equation to the given equation, . Because both equations represent the same line, the values for slope m are equal. Thus, a = 2. Like slope, the values for the y-intercept b areequal. So, 7a = b. Substitute a = 2 into this equation. Thus, b = 14.

CCSS ARGUMENTS Determine whether the following statements are sometimes, always, or never true.eSolutions Manual - Powered by Cognero


CCSS ARGUMENTS Determine whether the following statements are sometimes, always, or never true.Explain your reasoning.

��If f (x) and g(x) are inverse functions, then f (a) = b and g(b) = a.

62/87,21��Always; sample answer: If f (a) = b, then the graph of f (x) includes the point (a, b). If f (x) and g(x) are inverses, then the graph of g(x) includes the point (b, a). If (b, a) is included on the graph of g(x), then g(b) = a.

��If f (a) = b and g(b) = a, then f (x) and g(x) are inverse functions.

62/87,21��Sometimes; sample answer: f (x ) and g(x ) do not need to be inverse functions for f (a) = b and g(b) = a. For example, if f (x ) = 2x + 10, then f (2) = 14 and if g (x ) = x ± 12, then g (14) = 2, but f (x ) and g(x ) are not inverse functions. However, if f (x ) and g (x ) are inverse functions, then f (a) = b and g(b) = a.

��OPEN ENDED Give an example of a function and its inverse. Verify that the two functions are inverses by graphing the functions and the line y = x on the same coordinate plane.

62/87,21��6DPSOH�DQVZHU�� f(x) = 2x + 6, f -1(x ) = x ± 3

�

��WRITING IN MATH Explain why it may be helpful to find the inverse of a function.

62/87,21��Sample answer: A situation may require substituting values for the dependent variable into a function. By finding the inverse of the function, the dependent variable becomes the independent variable. This makes the substitution an easier process.

��Which equation represents a line that is perpendicular to the graph and passes through the point at (2, 0)?



��Which equation represents a line that is perpendicular to the graph and passes through the point at (2, 0)? �

� $� y = 3x í 6 � %� y = í3x + 6 �

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62/87,21��Find the slope of the line in the graph. The line passes through the points (±1, 1) and (0, 4). Find the slope of the line. �

The slope of the line shown in the graph is 3. So, a line that is perpendicular to the one shown in the graph will have a

slope of ± . Use the point slope formula to find the equation of the perpendicular line if it passes through the point

(2, 0). �

So, the equation that represents a line that is perpendicular to the line shown in the graph and passes through (2, 0)

is . So, the correct choice is C.

��A giant tortoise travels at a rate of 0.17 mile per hour. Which equation models the time t it would take the giant



��A giant tortoise travels at a rate of 0.17 mile per hour. Which equation models the time t it would take the giant tortoise to travel 0.8 mile? �

)��

� *� t = (0.17)(0.8) �

+��

�

-��

62/87,21��To find the time t it would take the tortoise to travel 0.8 mile, use the formula d = rt, where d = distance and r = rate.Then, solve for t.

So, the correct choice is F.

*(20(75<� JKL JNM a



��*(20(75<� If JKL is similar to JNM what is the value of a?

$� 62.5 � %� 105 � &��125 � '� 155.5

62/87,21��Triangle JMN is similar to triangle JLK because ӘM DQG�ӘL are right angles, ӘK and ӘM are marked congruent and ӘKJL and ӘMJN are vertical angles. Similar triangles congruent corresponding angles and proportional corresponding sides. � Use a proportion to find the value of a. �

� So, the correct choice is B.

*5,''('�5(63216(� x x x



��*5,''('�5(63216(� What is the difference in the value of 2.1(x + 3.2), when x = 5 and when x = 3?

62/87,21��Evaluate 2.1(x + 3.2), when x = 5.�

� Evaluate 2.1(x + 3.2), when x = 3.�

� Find the difference of the results. 17.22 ± 13.02 = 4.2 So, the difference is 4.2.

Write an equation of the regression line for the data in each table.

��

62/87,21��8VH�D�FDOFXODWRU�WR�ILQG�WKH�HTXDWLRQ�RI�WKH�UHJUHVVLRQ�OLQH�� Step 1: Enter the data into the lists by pressing and selecting the EDIT option. Enter the x values in List 1(L1) and the y values in List 2 (L2�� Step 2: Perform the regression by pressing and selecting the CALC option. Scroll down to LinReg(ax +

b) and press ��

� The equation of the regression line is y = 2.3x + 1.5.



��

62/87,21��Use a calculator to find the equation of the regression line. � Step 1: Enter the data into the lists by pressing and selecting the EDIT option. Enter the x values in List 1(L1) and the y values in List 2 (L2�� Step 2: Perform the regression by pressing and selecting the CALC option. Scroll down to v and press

��

� The equation of the regression line is y = 8.235x í 17.365.



��


b) and press ��




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b) and press ��


��7(676� Determine whether the graph below shows a positive , a negative, or no correlation. If there is a correlation, describe its meaning.

62/87,21��In general, the test scores increase as the amount of time spent studying increases. The regression line for this data has a positive slope and a correlation coefficient close to 0.5. Since 0.5 is not close to 1, the equation is not a good fit of the data. So, the graph shows a weak positive correlation. This means the more you study, the better your test score is likely to be.

Suppose y varies directly as x.



Suppose y varies directly as x.��If y = 2.5 when x = 0.5, find y when x = 20.

62/87,21��Find the constant of variation, k . �

� So, the direct variation equation is y = 5x.�

� So, y = 100 when x = 5.

��If y = í6.6 when x = 9.9, find y when x = 6.6.


So, the direct variation equation is y = x.

�

� So, y = ±4.4 when x = 6.6.

��If y = 2.6 when x = 0.25, find y when x = 1.125.



��If y = 2.6 when x = 0.25, find y when x = 1.125.


� So, the direct variation equation is y = 10.4x.�

� So, y = 11.7 when x = 1.125.

��If y = 6 when x = 0.6, find x when y = 12.


� So, the direct variation equation is y = 10x.�

� So, x = 1.2 when y = 12.

Solve each equation.��104 = k í 67

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x



��í4 + x = í7

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�� = í11

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�� = 14

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��í82 = 18 ± n

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�� = í27

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4%2d7 inverse linear functions c4 l7...write the final equation in slope -lqwhufhswirup so, ....

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