Composition & Inverses of Functions

Notes # Alpha 2

Operations with Functions: Assume f and g are functions. Sum:

(f + g)(x) = f(x) + g(x)

Difference:

(f – g)(x) = f(x) – g(x)

Product:

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

Quotient:

f f ( x) ( x) , g ( x) 0 g ( x) g

Ex A: Given f ( x)

x and g ( x) x3

#1)

f (x) g

#2)

(f + g)(x) =

x 2 2 x , find each function below. Note: When applicable, make sure you find the excluded values.

Linear Relations & Functions Page 1 of 4

Composition & Inverses of Functions Ex A: Given f ( x)

#3)

x and g ( x) x3

x 2 2 x , find each function below.

(f – g)(x) =

Composition of Functions: Given functions f and g, the composite function f ◦ g can be described by the following equation. [f ◦ g](x) = f(g(x)) The domain of f ◦ g includes all the elements x in the domain of g for which g(x) is in the domain of f. Ex B: Find [f ◦ g](x) and [g ◦ f](x).

#1)

f(x) = ½x – 5 g(x) = x + 7

#2)

f(x) = 3x2 g(x) = x – 2

[f ◦ g](x) =

[f ◦ g](x) =

[g ◦ f](x) =

[g ◦ f](x) =

Note: When doing composition, you are to perform the substitutions from right to left.

Linear Relations & Functions Page 2 of 4

Composition & Inverses of Functions Iteration: The composition of a function to itself.

Inverse functions:

Notes # Alpha 2

Ex: If f(x) = 2x + 4, find f(f(f(x))).

Two functions f and g are inverse functions iff [f ◦ g](x) = [g ◦ f](x) = x.

Ex C: Determine if the given functions are inverses of each other. Circle yes or no.

#1)

Note: Find [f ◦ g](x) and [g ◦ f](x). If they are both “x”, they are inverses of each other.

f(x) = 4x – 7 g(x) = x 7 4

Inverses? #2)

Yes

No

f(x) = x – 7 g(x) = x + 7

Inverses?

Yes

No Linear Relations & Functions Page 3 of 4

Composition & Inverses of Functions Property of Inverse Functions:

Suppose f and f -1 are inverse functions. Then, f(x) = y iff f -1(y) = x.

Ex D: Find the inverse of each function. Then decide whether the inverse is a function by circling yes or no.

#1)

f(x) = 4x + 4

#2)

f -1(x) = Function? #3)

Note: To find the inverse, substitute y for f(x). Then exchange all x’s and y’s. Solve the equation for y. Then sub f –1(x) for y.

f(x) = x3

f -1(x) = Yes

No

Function?

Yes

No

f(x) = x2 – 9

f -1(x) =

Function?

Yes

No Linear Relations & Functions Page 4 of 4