Function composition and function analysis Knowledge-Based Download PDF

Questions

1. 1.

   Let $f(x) = \sqrt{x + 5}$ and $g(x) = x^2 - 1$ be real-valued functions. What is the range of the composite function $(f \circ g)(x)$?

   • A) $[0, \infty)$
   • B) $[2, \infty)$
   • C) $[\sqrt{5}, \infty)$
   • D) $[4, \infty)$
2. 2.

   Let $f(x) = 3x + 4$ and $g(x) = 2x + k$ be real-valued functions, where $k$ is a real constant. If $(f \circ g)(x) = (g \circ f)(x)$ for all real numbers $x$, what is the value of $k$?

   • A) 2
   • B) 4
   • C) -2
   • D) 6
3. 3.

   Let $f(x) = x^2 - 4x + 1$ and $g(x) = 2x - 3$ be real-valued functions. What is the sum of all real solutions to the equation $(f \circ g)(x) = -2$?

   • A) 4
   • B) 5
   • C) 6
   • D) 7
4. 4.

   Given the real-valued functions $f(x) = \frac{1}{x - 3}$ and $g(x) = \sqrt{x + 2}$, what is the domain of the composite function $(f \circ g)(x)$?

   • A) $(-2, 7) \cup (7, \infty)$
   • B) $[-2, 7) \cup (7, \infty)$
   • C) $[3, 7) \cup (7, \infty)$
   • D) $[-2, \infty)$
5. 5.

   Let $f(x) = 2x - 5$ and $(f \circ g)(x) = 4x^2 - 6x + 3$. Which of the following expressions represents the function $g(x)$?

   • A) $g(x) = 2x^2 - 3x - 1$
   • B) $g(x) = 2x^2 - 3x + 4$
   • C) $g(x) = 4x^2 - 6x + 8$
   • D) $g(x) = 2x^2 - 6x + 4$