Question: Determining whether two functions are inverses of each other For each pair of functions \( f…

Determining whether two functions are inverses of each other

For each pair of functions \( f \) and \( g \) below, find \( f(g(x)) \) and \( g(f(x)) \). Then, determine whether \( f \) and \( g \) are inverses of each other.

Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.)

(a) \( f(x) = 2x - 5 \) \( g(x) = \frac{x + 5}{2} \)

\( f(g(x)) = \boxed{\phantom{xxxx}} \) \( g(f(x)) = \boxed{\phantom{xxxx}} \)

\( f \) and \( g \) are inverses of each other \( f \) and \( g \) are not inverses of each other

(b) \( f(x) = 2x \) \( g(x) = 2x \)

\( f(g(x)) = \boxed{\phantom{xxxx}} \) \( g(f(x)) = \boxed{\phantom{xxxx}} \)

\( f \) and \( g \) are inverses of each other \( f \) and \( g \) are not inverses of each other

Solution

For each pair of functions \( f \) and \( g \), we need to find \( f(g(x)) \) and \( g(f(x)) \). Then, determine whether \( f \) and \( g \) are inverses of each other. (a) Given: \[ f(x) = 2x - 5 \] \[ g(x) = \frac{x + 5}{2} \] 1. Find \( f(g(x)) \): Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x+5}{2}\right) = 2\left(\frac{x+5}{2}\right) - 5 \] Simplify: \[ = x + 5 - 5 = x \] 2. Find \( g(f(x)) \): Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(2x - 5) = \frac{(2x - 5) + 5}{2} \] Simplify: \[ = \frac{2x}{2} = x \] Since \( f(g(x)) = x \) and \( g(f(x)) = x \), \( f \) and \( g \) are inverses of each other. (b) Given: \[ f(x) = 2x \] \[ g(x) = 2x \] 1. Find \( f(g(x)) \): Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(2x) = 2(2x) = 4x \] 2. Find \( g(f(x)) \): Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(2x) = 2(2x) = 4x \] Since \( f(g(x)) = 4x \) and \( g(f(x)) = 4x \), \( f \) and \( g \) are not inverses of each other.