Function transformations and inverses Knowledge-Based Download PDF

Questions

1. 1.

   Let $f$ be an invertible function. Which geometric transformation maps the graph of $y = f(x)$ onto the graph of $y = -f^{-1}(-x)$?

   • A) Reflection across the line $y = x$
   • B) Reflection across the line $y = -x$
   • C) Reflection across the $y$-axis
   • D) A rotation of $180^\circ$ about the origin
2. 2.

   An invertible function $f$ has domain $[-5, 3]$ and range $[2, 10]$. What are the domain and range of the function $g(x) = 2f^{-1}(x + 4) - 3$?

   • A) Domain: $[-9, -1]$, Range: $[1, 17]$
   • B) Domain: $[6, 14]$, Range: $[-13, 3]$
   • C) Domain: $[-2, 6]$, Range: $[-13, 3]$
   • D) Domain: $[-2, 6]$, Range: $[1, 17]$
3. 3.

   Let $f$ be an invertible function, and let the function $g$ be defined by $g(x) = 2f(3x - 6) + 5$. Which of the following expressions represents the inverse function $g^{-1}(x)$?

   • A) $g^{-1}(x) = 3f^{-1}\left(\frac{x - 5}{2}\right) + 6$
   • B) $g^{-1}(x) = \frac{1}{3}f^{-1}\left(\frac{x - 5}{2}\right) + 2$
   • C) $g^{-1}(x) = \frac{1}{3}f^{-1}(2x - 5) + 2$
   • D) $g^{-1}(x) = \frac{1}{2}f^{-1}\left(\frac{x - 6}{3}\right) + 5$
4. 4.

   The graph of an invertible function $f$ contains the point $(2, -5)$. Which of the following points must lie on the graph of the function $g(x) = f^{-1}(x + 1) - 4$?

   • A) $(1, -9)$
   • B) $(-4, -2)$
   • C) $(-6, 6)$
   • D) $(-6, -2)$
5. 5.

   The function $f$ is defined by $f(x) = 3 - \sqrt{2x + 1}$ for $x \ge -\frac{1}{2}$. Which expression represents the inverse function $f^{-1}(x)$ along with its restricted domain?

   • A) $f^{-1}(x) = \frac{(x - 3)^2 + 1}{2}$ for $x \le 3$
   • B) $f^{-1}(x) = \frac{(x - 3)^2 - 1}{2}$ for $x \le 3$
   • C) $f^{-1}(x) = \frac{(x - 3)^2 - 1}{2}$ for $x \ge 3$
   • D) $f^{-1}(x) = \frac{(x + 3)^2 - 1}{2}$ for $x \ge -\frac{1}{2}$