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CCSS.Math.Content.HSF-IF.B.4, HSF-BF.B.45•
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5•2 Inverse Functions and Relations
Find Inverses
You have studied inverse operations such as multiplication and division. The inverse of a relation or function can be found algebraically. The graphs of inverse functions are reflections about the line y = x.
Inverse RelationsTwo relations are inverse relations if and only if whenever
one relation contains the element (a, b), the other
relation contains the element (b, a).
Inverse Functions
Two functions f and g are inverse functions if and only if
both [f ◦ g](x) and [g ◦ f ](x) are the identity function.
Find and Graph an Inverse
Find the inverse of the function f (x) = 2 _ 5 x - 1 _ 5 . Then graph
the function and its inverse.
Step 1: Replace f(x) with y in the original equation.
f(x) = 2 _ 5 x - 1 _ 5
y = 2 _ 5 x - 1 _ 5
Step 2: Interchange x and y.
x = 2 _ 5 y - 1 _ 5
Step 3: Solve for y.
x = 2 _ 5 y - 1 _ 5 5x = 2y - 1 Multiply each side by 5.
5x + 1 = 2y Add 1 to each side.
1 _ 2 (5x + 1) = y Divide each side by 2.
5 _ 2 x + 1 _ 2 = y Distribute.
The inverse of f(x) = 2 _ 5 x - 1 _ 5 is f -1(x) = 5 _ 2 x + 1 _ 2 .
xO
f (x) = 2–5x - 1–5
2 4
2
-2
-2
-4
-4
4f ( x )
f –1(x) = 5–2x + 1–2
EXAMPLE
Program: FL MATH REPRINT Component: HANDBOOK1st Pass
Vendor: LASERWORDS Grade: ALGEBRA 2
170 HotTopic 5
170-172_ALG2_MS_S_HB_C05_L02_142916.indd 170170-172_ALG2_MS_S_HB_C05_L02_142916.indd 170 25/10/13 3:53 PM25/10/13 3:53 PM
Verifying Inverses
Using composite functions, it is possible to determine whether two given functions are inverses.
If both compositions equal the identity function I(x) = x, then the functions are inverse functions.
Inverse FunctionsTwo functions f(x) and g(x) are inverse functions if and
only if [f ◦ g](x) = x and [g ◦ f](x) = x.
Verify that Two Functions are Inverses
Determine whether the functions are inverses.a. f(x) = 2x - 7 and g(x) = 1 _ 2 (x + 7)
[ f ◦ g](x) = f[ g(x)] [ g ◦ f ](x) = g[ f(x)]
= f [ 1 _ 2 (x + 7)] = g(2x - 7)
= 2 [ 1 _ 2 (x + 7)] - 7 = 1 _ 2 (2x - 7 + 7)
= x + 7 - 7 = x
= x
The functions are inverses since both [ f ◦ g](x) = x and [ g ◦ f ](x) = x.
b. f(x) = 4x + 1 _ 3 and g(x) = 1 _ 4 x - 3
[ f ◦ g](x) = f[ g(x)]
= f ( 1 _ 4 x - 3)
= 4 ( 1 _ 4 x - 3) + 1 _ 3
= x - 12 + 1 _ 3
= x - 11 2 _ 3
Since [ f ◦ g](x) ≠ x, the functions are not inverses.
EXAMPLE
Program: FL MATH REPRINT Component: HANDBOOK1st Pass
Vendor: LASERWORDS Grade: ALGEBRA 2
Inverse Functions and Relations 171
170-172_ALG2_FL_S_HB_C05_L02_144034.indd 171170-172_ALG2_FL_S_HB_C05_L02_144034.indd 171 16/08/13 4:23 PM16/08/13 4:23 PM
5•2
EX
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5•2 ExercisesFind the inverse of each function. Then graph the function and its inverse.
1. f(x) = 2 _ 3 x - 1 2. f(x) = 2x - 3
Determine whether each pair of functions are inverse functions. Write yes or no.
3. f(x) = 3x - 1 4. f(x) = 1 _ 4 x + 5
g(x) = 1 _ 3 x + 1 _ 3 g(x) = 4x - 20
5. f(x) = 1 _ 2 x - 10 6. f(x) = 2x + 5
g(x) = 2x + 1 _ 10 g(x) = 5x + 2
7. f(x) = 8x - 12 8. f(x) = -2x + 3
g(x) = 1 _ 8 x + 12 g(x) = - 1 _ 2 x + 3 _ 2
9. f(x) = 4x - 1 _ 2 10. f(x) = 2x - 3 _ 5
g(x) = 1 _ 4 x + 1 _ 8 g(x) = 1 _ 10 (5x + 3)
11. f(x) = 4x + 1 _ 2 12. f(x) = 10 - x _ 2
g(x) = 1 _ 2 x - 3 _ 2 g(x) = 20 - 2x
13. f(x) = 4x - 4 _ 5 14. f(x) = 9 + 3 _ 2 x
g(x) = x _ 4
+ 1 _ 5 g(x) = 2 _ 3 x - 6
15. EXERCISE Alex began a new exercise routine. To gain the maximum benefit from his exercise, Alex calculated his maximum target heart rate using the function f(x) = 0.85(220 - x), where x represents his age. Find the invers e of this function.
Program: FL MATH Component: HANDBOOKPDF Pass
Vendor: LASERWORDS Grade: ALGEBRA 2
172 HotTopic 5
170-172_ALG2_FL_S_HB_C05_L02_144034.indd 172170-172_ALG2_FL_S_HB_C05_L02_144034.indd 172 05/04/13 2:03 AM05/04/13 2:03 AM