Function composition and function analysis Knowledge-Based Practice

Questions

1. 1.

   Given the real-valued function $f(x) = \frac{x}{\sqrt{1 + x^2}}$, which expression represents $(f \circ f \circ f)(x)$?

   • A) $\frac{x}{\sqrt{1 + 2x^2}}$
   • B) $\frac{x}{\sqrt{1 + 3x^2}}$
   • C) $\frac{x}{\sqrt{1 + 9x^2}}$
   • D) $\frac{3x}{\sqrt{1 + x^2}}$
2. 2.

   For the real-valued functions $f(x) = x^2 - 6x + 2$ and $g(x) = \sqrt{x} + 4$ defined for $x \ge 0$, what is the range of the composite function $(f \circ g)(x)$?

   • A) $[-6, \infty)$
   • B) $[-7, \infty)$
   • C) $[2, \infty)$
   • D) $[10, \infty)$
3. 3.

   Let $f$ and $g$ be real-valued functions such that $g(x) = 2x - 1$ and $(f \circ g)(x) = 4x^2 - 6x + 5$. Which of the following defines $f(x)$?

   • A) $f(x) = x^2 - x + 3$
   • B) $f(x) = x^2 + x + 3$
   • C) $f(x) = 2x^2 - 3x + 5$
   • D) $f(x) = x^2 - 5x + 9$
4. 4.

   For the real-valued functions $f(x) = \sqrt{25 - x^2}$ and $g(x) = |2x - 1|$, what is the domain of the composite function $(f \circ g)(x)$?

   • A) $[-3, 2]$
   • B) $[-2, 3]$
   • C) $[-5, 5]$
   • D) $\left[-\frac{5}{2}, \frac{5}{2}\right]$
5. 5.

   Given the real-valued functions $f(x) = 3x - 4$ and $g(x) = \frac{2x + 5}{x - 1}$ defined for $x \neq 1$, find the value of $x$ that satisfies $(f \circ g)(x) = 5$.

   • A) 2
   • B) 4
   • C) 6
   • D) 8

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