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

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

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 −9 + 9 4 = 4𝑥 4 = x g 𝑥+9 4 = 4 𝑥+9 4 – 9 g(f(x)) = = x + 9 – 9 = x INVERSES

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) = 2 1 2 𝑥+9 – 9 = x + 18 – 9 = x + 9 NOT INVERSES

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

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 + 5 3 𝑓 −1 (x) = − 2 3 𝑥 + 5 3

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