Pre-Calculus
Core Problems

Concept: The inverse of a function swaps input and output. To find the domain of \(f^{-1}(x)\), note it equals the range of \(f(x)\).

If \(f(x) = \sqrt{x - 3} + 2\), which of the following is the correct expression for \(f^{-1}(x)\)?

Concept: \((f \circ g)(x) = f(g(x))\). Substitute \(g(x)\) into \(f\).

If \(f(x) = 2x^2 - 1\) and \(g(x) = x + 3\), what is \((f \circ g)(x)\)?

Concept: Even: \(f(-x)=f(x)\) (y-axis symmetry). Odd: \(f(-x)=-f(x)\) (origin symmetry).

Which function is odd?

Concept: The Remainder Theorem states: when \(p(x)\) is divided by \((x-c)\), the remainder is \(p(c)\).

What is the remainder when \(p(x) = 2x^3 - x^2 + 3x - 1\) is divided by \((x - 2)\)?

Concept: Factor Theorem: \((x-a)\) is a factor iff \(f(a)=0\). Rational Root Test: possible rational roots = \(\pm\frac{p}{q}\) where \(p\mid\text{constant},\; q\mid\text{leading coeff}\).

Given \(f(x) = x^3 - 6x^2 + 11x - 6\), which set lists all real roots?

Concept: Vertical asymptotes occur where the denominator = 0 (and numerator ≠ 0). Horizontal asymptote: if deg(num) = deg(den), HA = ratio of leading coefficients.

For \(f(x) = \dfrac{3x^2 - 1}{2x^2 + 5}\), identify the horizontal asymptote.

Concept: A hole occurs when a factor cancels from both numerator and denominator. A vertical asymptote occurs at the remaining denominator zeros.

\(f(x) = \dfrac{x^2 - 4}{x^2 - x - 2}\). Does the graph have a hole, a vertical asymptote, or both?

Key rules: \(\log_b(MN)=\log_b M+\log_b N\), \(\log_b\!\left(\frac{M}{N}\right)=\log_b M - \log_b N\), \(\log_b(M^k)=k\log_b M\).

Simplify: \(\log_2 96 - \log_2 3\)

Concept: To solve \(a^{f(x)}=a^{g(x)}\), set exponents equal. If bases differ, take logarithm of both sides.

Solve for \(x\): \(4^{x+1} = 8^{x-1}\)

Formula: \(\log_b a = \dfrac{\ln a}{\ln b} = \dfrac{\log a}{\log b}\).

Using \(\log 2 \approx 0.301\) and \(\log 3 \approx 0.477\), evaluate \(\log_9 8\) to the nearest hundredth.

Memorize: \(\sin 30°=\frac{1}{2}\), \(\cos 30°=\frac{\sqrt{3}}{2}\), \(\sin 45°=\frac{\sqrt{2}}{2}\), \(\sin 60°=\frac{\sqrt{3}}{2}\).

What is the exact value of \(\sin\!\left(\dfrac{7\pi}{6}\right)\)?

For \(y = A\sin(Bx - C) + D\): Amplitude = \(|A|\), Period = \(\frac{2\pi}{|B|}\), Phase Shift = \(\frac{C}{B}\), Vertical Shift = \(D\).

For \(y = 3\sin\!\left(2x - \dfrac{\pi}{3}\right) + 1\), what is the phase shift?

Identities: \(\sin^2\theta+\cos^2\theta=1\), \(1+\tan^2\theta=\sec^2\theta\), \(1+\cot^2\theta=\csc^2\theta\).

If \(\cos\theta = -\dfrac{3}{5}\) and \(\theta\) is in Quadrant II, what is \(\tan\theta\)?

Formula: \(\sin(2\theta) = 2\sin\theta\cos\theta\), \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\).

If \(\sin\theta = \dfrac{5}{13}\) and \(\theta\) is in Quadrant I, find \(\sin(2\theta)\).

Formula: \(S_n = \dfrac{n}{2}(a_1 + a_n)\) or \(S_n = \dfrac{n}{2}[2a_1 + (n-1)d]\).

Find the sum of the first 20 terms of the arithmetic sequence: \(3, 7, 11, 15, \ldots\)

Formula: \(S_\infty = \dfrac{a_1}{1-r}\), valid when \(|r| < 1\).

Find the sum of the infinite geometric series: \(16 + 8 + 4 + 2 + \cdots\)

Standard form of ellipse: \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\). If \(a>b\), major axis is horizontal; if \(b>a\), vertical.

For \(\dfrac{(x-1)^2}{25}+\dfrac{(y+2)^2}{9}=1\), what are the coordinates of the center and the length of the major axis?

Standard form: \((x-h)^2 = 4p(y-k)\). Focus at \((h, k+p)\). Directrix: \(y = k-p\).

For the parabola \((x-2)^2 = 8(y+1)\), what is the focus?

Formula: \(\vec{u}\cdot\vec{v} = |\vec{u}||\vec{v}|\cos\theta\), so \(\cos\theta = \dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\).

Let \(\vec{u} = \langle 3, 4 \rangle\) and \(\vec{v} = \langle 0, 1 \rangle\). What is the angle between \(\vec{u}\) and \(\vec{v}\)?

Concept: When direct substitution gives \(\frac{0}{0}\), factor and cancel.

Evaluate: \(\displaystyle\lim_{x \to 3}\;\dfrac{x^2 - 9}{x - 3}\)