Using Composition to Test If Two Functions Are Inverses

Q2. Use composition to decide if two functions are inverses of each other.

Background

Topic: Functions and Inverses

This question tests your understanding of function composition and how to verify if two functions are inverses. Two functions, $f(x)$ and $f^{-1}(x)$, are inverses if and only if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in their domains.

Key Terms and Formulas

• Function Composition: $(f \circ g)(x) = f(g(x))$

• Inverse Function: $f^{-1}(x)$ is the function that "undoes" $f(x)$.

• Inverse Test: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Step-by-Step Guidance (Part a)

1. Write down the given functions: $f(x) = \frac{x-2}{5}$ and $f^{-1}(x) = 5x + 2$.

2. Set up the first composition: $f(f^{-1}(x))$. Substitute $f^{-1}(x)$ into $f(x)$:

   $f(f^{-1}(x)) = f(5x + 2) = \frac{(5x + 2) - 2}{5}$

3. Simplify the numerator inside the fraction.

4. Set up the second composition: $f^{-1}(f(x))$. Substitute $f(x)$ into $f^{-1}(x)$:

   $f^{-1}(f(x)) = 5\left(\frac{x-2}{5}\right) + 2$

5. Simplify the expression inside the parentheses and then add 2.

Step-by-Step Guidance (Part b)

1. Write down the given functions: $f(x) = 2\sqrt{x+3} - 2$ and $f^{-1}(x) = \left(\frac{x+2}{2}\right)^2 - 3$.

2. Set up the first composition: $f(f^{-1}(x))$. Substitute $f^{-1}(x)$ into $f(x)$:

   $f(f^{-1}(x)) = 2\sqrt{\left(\frac{x+2}{2}\right)^2 - 3 + 3} - 2$

3. Simplify the expression under the square root and then continue simplifying the composition.

4. Set up the second composition: $f^{-1}(f(x))$. Substitute $f(x)$ into $f^{-1}(x)$:

   $f^{-1}(f(x)) = \left(\frac{2\sqrt{x+3} - 2 + 2}{2}\right)^2 - 3$

5. Simplify the numerator and then the entire expression inside the parentheses before squaring and subtracting 3.

Try solving on your own before revealing the answer!