Problem 6 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7}$$

Step-by-Step Solution

Answer

The functions \( f(x) = 3x-7 \) and \( g(x) = \frac{x+3}{7} \) are inverses of each other because both composite functions \( f(g(x)) \) and \( g(f(x)) \) yield \( x \).

Step 1: Find \( f(g(x)) \)

Replace \( x \) in \( f(x) \) with \( g(x) \) to get: \n\n \( f(g(x)) = f \left( \frac{x+3}{7} \right) = 3 \left( \frac{x+3}{7} \right) -7. \) Do the calculation to simplify the expression.

Step 2: Simplify \( f(g(x)) \)

Simplify \n\n \( 3 \left( \frac{x+3}{7} \right) -7 \) to get \( x \). Therefore, \( f(g(x)) = x . \)

Step 3: Find \( g(f(x)) \)

Replace \( x \) in \( g(x) \) with \( f(x) \) to get: \n\n \( g(f(x)) = g(3x-7) = \frac{3x-7+3}{7} . \) Do the calculation to simplify the expression.

Step 4: Simplify \( g(f(x)) \)

Simplify the expression \n\n \( \frac{3x-7+3}{7} \) to get \( x \). Therefore, \( g(f(x)) = x . \)

Step 5: Identify if the Functions are Inverses

Because both composite functions \( f(g(x)) \) and \( g(f(x)) \) yield \( x \), therefore \( f(x) \) and \( g(x) \) are inverses of each other.