1. Warm Up Determine the domain of f(g(x)). f(x) = g(x) =
2𝑥 −10
x + 2
2 𝑥+2 −10
f(g(x)) =
f(x + 2) =
= 2𝑥+4−10
= 2𝑥−6
2x – 6 ≥ 0
all real numbers where x ≥ 3

2. Inverse Functions if and only if… (f о g)(x) = (g о f)(x) = x
Two functions f and g are inverses
if and only if… 1. 2.
Graphs are symmetrical over y = x
(f о g)(x)
= (g о f)(x)
= x
𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥 =𝑥

3. Determine if the given functions are inverses of one another.
Examples
Determine if the given functions are inverses of one another.
1. f(x) = 𝒙+𝟗 𝟒
g(x) = 4x – 9
f(g(x)) =
f(4x – 9)
= 4𝑥 −
= 4𝑥 4
= x
g 𝑥+9 4
= 4 𝑥 – 9
g(f(x)) =
= x + 9 – 9
= x
INVERSES

4. f(g(x)) = f( 1 2 x + 9) = 2 1 2 𝑥+9 – 9 = x + 18 – 9 = x + 9
2. f(x) = 2x - 9 g(x) = 𝟏 𝟐 x + 9
f(g(x)) =
f( 1 2 x + 9)
= 𝑥+9 – 9
= x + 18 – 9
= x + 9
NOT INVERSES

5. To Find the Inverse of a single function…
Switch the x and y values.
Solve for y.
Replace y with f -1(x).

6. y = 5−3𝑥 2 x = 5−3𝑦 2 2x = 5 – 3y –3y = 2x – 5 2x – 5 = –3y
Examples
Determine the inverse of f(x)
1. f(x) = 𝟓 −𝟑𝒙 𝟐
y = 5−3𝑥 2
x = 5−3𝑦 2
2x = 5 – 3y
–3y = 2x – 5
2x – 5 = –3y
y = − 2 3 x
𝑓 −1 (x) = − 2 3 𝑥

7. y = 𝑥 2 – 1 x = 𝑦 2 – 1 x + 1 = 𝑦 2 𝑦 2 = x + 1 y = ± 𝑥+1
2. f(x) = x² – 1
y = 𝑥 2 – 1
x = 𝑦 2 – 1
x + 1 = 𝑦 2
𝑦 2 = x + 1
y = ± 𝑥+1
𝑓 −1 (x) = ± 𝑥+1