Chapter 2 — Self-Assessment Quiz

10 questions, ~20 minutes. Answer before expanding the solution. Aim for 7/10. Each answer cites the section to review.

1. A function is best defined as:

  • A) A formula involving $x$ and $y$
  • B) A rule that assigns a unique output to each input
  • C) A graph in the $xy$-plane
  • D) An equation with one variable
Answer **B.** A function is a *rule*. A formula and a graph are *representations* of the rule, not the rule itself. *Reference: §2.1.*

2. The natural domain of $f(x) = \dfrac{\sqrt{x - 1}}{x - 3}$ is:

  • A) $\mathbb{R}$
  • B) $[1, \infty)$
  • C) $[1, 3) \cup (3, \infty)$
  • D) $(3, \infty)$
Answer **C.** The numerator requires $x - 1 \ge 0$, i.e. $x \ge 1$; the denominator requires $x \ne 3$. Intersecting gives $[1, 3) \cup (3, \infty)$. *Reference: §2.1 (finding the natural domain).*

3. The graph of $y = (x - 2)^2 + 3$ is obtained from $y = x^2$ by:

  • A) Shifting right $2$, up $3$
  • B) Shifting left $2$, up $3$
  • C) Shifting right $2$, down $3$
  • D) Reflecting across the $x$-axis
Answer **A.** Replacing $x$ by $x - 2$ shifts the graph *right* (inside-the-function signs run backwards); adding $3$ outside shifts up. *Reference: §2.3.*

4. Which function is its own inverse?

  • A) $f(x) = x^2$
  • B) $f(x) = 2x$
  • C) $f(x) = 1/x$
  • D) $f(x) = e^x$
Answer **C.** $f(f(x)) = f(1/x) = 1/(1/x) = x$, so $f$ undoes itself. *Reference: §2.5.*

5. $\sin(\pi/3) = $

  • A) $1/2$
  • B) $\sqrt{2}/2$
  • C) $\sqrt{3}/2$
  • D) $1$
Answer **C.** $\sin(\pi/3) = \sqrt{3}/2$, from the special-values table. *Reference: §2.2 — memorize this table.*

6. For $f(x) = e^x$, $f(\ln 5) = $

  • A) $5$
  • B) $e^5$
  • C) $\ln 5$
  • D) $1/5$
Answer **A.** $e^{\ln a} = a$ because $e^x$ and $\ln x$ are inverses. *Reference: §2.2 (logarithmic functions); inverses §2.5.*

7. For $f(x) = x^2$ and $g(x) = x + 1$, $(f \circ g)(2) = $

  • A) $5$
  • B) $9$
  • C) $7$
  • D) $4$
Answer **B.** $(f \circ g)(2) = f(g(2)) = f(3) = 9$. Apply $g$ first, then $f$. *Reference: §2.5 (composition).*

8. Which of these is a piecewise function?

  • A) $f(x) = |x|$
  • B) $f(x) = x^2 + 1$
  • C) $f(x) = \sin x$
  • D) $f(x) = 2^x$
Answer **A.** $|x| = x$ for $x \ge 0$ and $|x| = -x$ for $x < 0$ — two formulas on two parts of the domain. *Reference: §2.4.*

9. A bacterial population grows as $P(t) = 100 \cdot 3^t$ ($t$ in hours). When does it reach $8100$?

  • A) $t = 3$
  • B) $t = 4$
  • C) $t = \log_3 81$
  • D) Both B and C
Answer **D.** $8100/100 = 81 = 3^4$, so $t = 4$; and $\log_3 81 = 4$, so B and C name the same number. *Reference: §2.7 (modeling viewpoint).*

10. In calculus, angles are measured in:

  • A) Degrees
  • B) Radians
  • C) Either, as convenient
  • D) Gradians
Answer **B.** Radians. The clean identities $\frac{d}{dx}\sin x = \cos x$ and $\int \sin x\,dx = -\cos x + C$ hold only in radians; degrees inject a factor of $\pi/180$. *Reference: §2.2 (common pitfall).*

Scoring Guide

Score Interpretation Next step
9–10 Excellent — function fluency is solid. Proceed to Chapter 3 (The Limit).
7–8 Solid. Move on; review §2.3 and §2.5 if you missed those items.
5–6 Shaky foundations. Re-read the chapter and redo Parts A–B of the exercises.
Below 5 Precalculus gaps. Work through Appendix A — Precalculus Review before continuing.