Composition & Inverses of Functions
Notes # Alpha 2
Operations with Functions: Assume f and g are functions. Sum: Product: f g
(f + g)(x) = f(x) + g(x) (f • g)(x) = f(x) • g(x)
Difference: Quotient:
(f – g)(x) = f(x) – g(x)
f ( x) ( x ) = g ( x ) , g ( x ) ≠ 0
Ex A: Given f ( x ) =
#1)
f g
( x) =
#2)
(f + g)(x) =
x and g ( x ) = x 2 − 2 x , find each function below. x +3
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 ) = x 2 − 2 x , find each function below. x +3
(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)
g(x) =
Inverses? #2)
Note: Find [f ◦ g](x) and [g ◦ f](x). If they are both “x”, they are inverses of each other.
f(x) = 4x – 7
x +7 4
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