Function transformations and inverses Knowledge-Based Practice

Questions

1. 1.

   Let $f$ be an invertible function with domain $[2, 8]$ and range $[-1, 5]$. If $g(x) = 2f^{-1}(3 - x) + 1$, what are the domain and range of $g$?

   • A) Domain: $[-5, 1]$, Range: $[5, 17]$
   • B) Domain: $[-2, 4]$, Range: $[-1, 11]$
   • C) Domain: $[1, 7]$, Range: $[4, 16]$
   • D) Domain: $[-2, 4]$, Range: $[5, 17]$
2. 2.

   Let $f$ be an invertible function and define $g(x) = 5 - f^{-1}(2x)$. Which of the following expressions represents $g^{-1}(x)$ in terms of $f$?

   • A) $g^{-1}(x) = 2f(5 - x)$
   • B) $g^{-1}(x) = \frac{1}{2}f(5 - x)$
   • C) $g^{-1}(x) = \frac{1}{2}f(x - 5)$
   • D) $g^{-1}(x) = 5 - \frac{1}{2}f(x)$
3. 3.

   The graph of the function $g$ is obtained by reflecting the graph of $y = f(3x - 12)$ across the line $y = x$, where $f$ is an invertible function. Which of the following gives $g(x)$ in terms of $f^{-1}(x)$?

   • A) $g(x) = f^{-1}\left(\frac{1}{3}x + 4\right)$
   • B) $g(x) = 3f^{-1}(x) - 12$
   • C) $g(x) = \frac{1}{3}f^{-1}(x) - 4$
   • D) $g(x) = \frac{1}{3}f^{-1}(x) + 4$
4. 4.

   Let $f$ be an invertible function whose graph contains the point $(3, -2)$. If $g(x) = 2f(x + 1) - 4$, which ordered pair must lie on the graph of $y = g^{-1}(x)$?

   • A) $(-8, 2)$
   • B) $(2, -8)$
   • C) $(-8, 4)$
   • D) $(-4, 2)$
5. 5.

   Given the function $f(x) = 4x - 7$, let $g(x) = f\left(\frac{x+1}{3}\right) + 2$. What is the formula for the inverse function $g^{-1}(x)$?

   • A) $g^{-1}(x) = \frac{3x - 11}{4}$
   • B) $g^{-1}(x) = \frac{4x + 11}{3}$
   • C) $g^{-1}(x) = \frac{3x + 5}{4}$
   • D) $g^{-1}(x) = \frac{3x + 11}{4}$

Answers and step-by-step solutions appear after you finish the test.