SATProblem of the Day

2.5 Inverses of Functions2.5 Inverses of Functions2.5 Inverses of Functions2.5 Inverses of FunctionsObjectives: •Find the inverse of a relation or function•Determine whether the inverse of a function is a function

Example 1Solve the equation v = 50 + 3t for t.

1 50

t v3 3

v = 50 + 3t-50 -50

v - 50 = 3t 3 3

Inverse of a Relation

The domain of the inverse is the range of the original relation.

The range of the inverse is the domain of the original relation.

The inverse of a relation consisting of the ordered pairs (x, y) is the set of all ordered pairs (y, x).

Example 2Find the inverse of each relation. State whether the relation is a function. State whether the inverse is a function.

a) {(1,2), (4,-2), (3,2)}inverse: {(2,1), (-2,4), (2,3)}

functionnot a function

b) {(-2,4), (3,-4), (-8,-5)}inverse: {(4,-2), (-4,3), (-5,-

8)}

functionfunction

Example 3Find an equation for the inverse of .

2x 3y

5

2y 3x

5

2y 3

x5

5 5

interchange x and ysolve for y

5x 2y 3 solve for y-3 -35x - 3 = 2y

2 2

5 3

y x2 2

PracticeFind an equation for the inverse of

f(x) = 4x – 5.

Activity1) Graph y = 2x – 1.2) Graph y = x.

3) Graph the inverse of y = 2x – 1.

4) Graph y = -2x + 55) Graph y = x.

6) Graph the inverse of y = -2x + 5.

Graphs of Inverse Functions

The graph of the inverse of a function is the reflection of the graph of the function across the line y = x.

Horizontal-Line TestThe inverse of a function is a function iff every horizontal line intersects the graph of the given function at no more than one point.

Horizontal-Line Test

not a function function

Composition and Inverses

If f and g are functions and (f ○ g)(x) = (g ○ f)(x) = x then f and g are inverses of one another.

Example 4Show that f(x) = 4x – 3 and are inverses of each other.

1 3

g(x) x4 4

(f g)(x) f(g(x))

1 3x

4 4f

1 34 x 3

4 4

x 3 3

(g f )(x) g(f(x))

g(4x 3)

1 3

(4x 3)4 4

3 3

x4 4

x x

Since , the two functions are inverses of each other.

(f g)(x) (g f )(x) x

PracticeShow that f(x) = -5x + 7 and are inverses of each other.

1 7

g(x) x5 5

Homework

More Problems on Inverses

Collins WritingType 1

Compare and contrast the vertical- and horizontal-line tests.