Part III Review

How to use this review

This review mixes the whole Functions part of the course with no chapter labels, just like the Functions questions on the real ACT. Twenty questions test recognition across function notation, composition, domain, rate of change, zeros, end behavior, sequences, asymptotes, exponential growth, logarithms, transformations, piecewise functions, and inverses. A short addendum adds extra Sequences and Series and Logarithms questions, plus two trigonometry questions on right-triangle ratios and the Pythagorean identity. No calculator is required, though Desmos or a handheld may still help on a few.

Recognition checklist

Before you calculate, name what the question is testing. That choice usually decides the method.

• Function notation and composition: work from the inside out. $f(g(x))$ means evaluate $g$ first, then feed the result into $f$.
• Domain: a square root needs a non-negative radicand, and a denominator can never be zero. Both restrictions apply at once.
• Zeros: factor, then set each factor to zero. The factor $(x+7)$ gives the zero $x = -7$.
• End behavior: look at the degree (even or odd) and the sign of the leading coefficient.
• Rational functions: vertical asymptotes come from zeros of the denominator; for equal degrees, the horizontal asymptote is the ratio of leading coefficients.
• Sequences and series: $a_n = a_1 + (n-1)d$ or $a_1 r^{n-1}$ for a term. For a sum, use $\frac{n}{2}(a_1 + a_n)$ or $\frac{a_1(1-r^n)}{1-r}$.
• Logarithms: $\log_b(x) = k$ means $b^k = x$. Adding logs multiplies arguments, and subtracting logs divides them.
• Transformations: in $-2f(x-3)+1$, the $-2$ reflects and stretches, $x - 3$ shifts right 3, and $+1$ shifts up 1.
• Inverses: swap $x$ and $y$, solve, or ask which input gives the target output. Respect any domain restriction when choosing the sign of a root.

Strategy

Do the questions you recognize first, and use a quick check on each answer: substitute back into the original statement whenever it takes only a few seconds. Many wrong answers on this review come from a single predictable slip such as stopping one step early, dropping a sign, or reporting an intermediate value, so reread what the question actually asks for before you bubble in a letter.