HSF.BF.B.4.b Tutor Intake — Verifying Inverses by Composition

Complete the formal definition: Two functions $f$ and $g$ are inverses of each other
if and only if $f(g(x)) = $ [result1] for all $x$ in the domain of $g$,
AND $g(f(x)) = $ [result2] for all $x$ in the domain of $f$.

Consider $f(x) = x^2$ (with domain all real numbers) and $g(x) = \sqrt{x}$.

Which statement best explains why $f$ and $g$ are NOT inverses over all real numbers?

Let $f(x) = 3x - 6$ and $g(x) = \dfrac{x + 6}{3}$.

Compute $f(g(x))$ and simplify completely. What is the result?

Let $f(x) = \dfrac{x + 3}{5}$ and $g(x) = 5x - 3$.

Compute $g(f(x))$ and simplify. What is the result?

A student claims that $g(x) = \dfrac{x - 3}{4}$ is the inverse of $f(x) = 4x - 3$.

Compute $f(g(x))$. What does this reveal?

Let $f(x) = 2x + 1$ and $g(x) = \dfrac{x - 1}{2}$.

A student performs a numeric check with $a = 3$:

• $f(g(3)) = f(1) = 3$ ✓
• $g(f(3)) = g(7) = 3$ ✓

What can the student conclude from this numeric check?

Let $f(x) = \dfrac{x - 4}{2}$ and $g(x) = 2x + 4$.

Compute $g(f(10))$.

Enter your answer as a number.