CCSS.Math.Content.HSF-IF.B.4, HSF-BF.B.4 5• 2 INVERSE FUNCTIONS AND RELATIONS 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 Relations Two 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 . x O f (x) = 2 – 5 x - 1 – 5 2 4 2 -2 -2 -4 -4 4 f ( x ) f –1 (x) = 5 – 2 x + 1 – 2 EXAMPLE 170 HotTopic 5
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CCSS.Math.Content.HSF-IF.B.4, HSF-BF.B.4 5•2 InvereFus ... · CCSS.Math.Content.HSF-IF.B.4, HSF-BF.B.4 5 • 2 INVERSE FUNCTIONS AND RELATIONS 5•2 InvereFus nction s and Relations
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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
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
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