Pre-Calculus Master Series | 20 Essential Questions

1

Functions & Their Properties

Domain, Range, Composition, Inverse

\((f \circ g)(x) = f(g(x))\)
\(f^{-1}\) exists iff \(f\) is one-to-one (passes Horizontal Line Test)

Domain: all valid inputs; Range: all possible outputs
For \(f^{-1}\): swap \(x \leftrightarrow y\), solve for \(y\)
Even: \(f(-x)=f(x)\); Odd: \(f(-x)=-f(x)\)

Example

If \(f(x)=2x+3\), find \(f^{-1}(x)\).

Answer: \(y = 2x+3 \Rightarrow x = 2y+3 \Rightarrow f^{-1}(x) = \dfrac{x-3}{2}\)

2

Polynomials & Rational Functions

Zeros, Remainder Theorem, Asymptotes

Remainder Theorem: \(p(a)\) = remainder when \(p(x)\div(x-a)\)
Factor Theorem: \((x-a)\) is a factor iff \(p(a)=0\)

Rational zero candidates: \(\pm\dfrac{\text{factors of constant}}{\text{factors of leading coeff}}\)
Vertical asymptote: set denominator \(=0\) (cancel common factors first)
Horizontal asymptote: compare degrees of numerator & denominator

Example

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

Answer: \(p(2) = 8-4+1 = 5\)

3

Exponential & Logarithmic Functions

Log Laws, Natural Log, Solving Equations

\(\log_b(xy)=\log_b x+\log_b y\)
\(\log_b\!\left(\tfrac{x}{y}\right)=\log_b x-\log_b y\)
\(\log_b(x^n)=n\log_b x\)
Change of base: \(\log_b x = \dfrac{\ln x}{\ln b}\)

\(y=b^x \Leftrightarrow x=\log_b y\) (inverse relationship)
\(e^{\ln x}=x\) and \(\ln(e^x)=x\)

Example

Solve \(2^{x+1}=32\).

Answer: \(2^{x+1}=2^5 \Rightarrow x+1=5 \Rightarrow x=4\)

4

Trigonometry

Unit Circle, Identities, Graphs

\(\sin^2\theta+\cos^2\theta=1\)
\(\sin(2\theta)=2\sin\theta\cos\theta\)
\(\cos(2\theta)=\cos^2\theta-\sin^2\theta\)
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\)

Unit circle: \(\sin\) = y-coord, \(\cos\) = x-coord
Period of \(\sin(bx)\): \(T=\dfrac{2\pi}{|b|}\); Amplitude = \(|A|\)
Reference angles: Quadrant II: \(\pi-\theta\), III: \(\theta-\pi\), IV: \(2\pi-\theta\)

Example

Find \(\sin(150°)\).

Answer: \(150°\) is in Q2; ref angle \(=30°\); \(\sin(150°)=\sin(30°)=\tfrac{1}{2}\)

5

Conic Sections

Circle, Parabola, Ellipse, Hyperbola

Circle: \((x-h)^2+(y-k)^2=r^2\)
Parabola: \(x^2=4py\) (opens up if \(p>0\))
Ellipse: \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), \(a>b>0\)
Hyperbola: \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\)

Example

Center and radius of \((x-3)^2+(y+2)^2=25\)?

Answer: Center \((3,-2)\), radius \(r=5\)

6

Sequences & Series

Arithmetic, Geometric, Sigma Notation

Arithmetic: \(a_n=a_1+(n-1)d\), \(S_n=\dfrac{n}{2}(a_1+a_n)\)
Geometric: \(a_n=a_1 r^{n-1}\), \(S_n=\dfrac{a_1(1-r^n)}{1-r}\)
Infinite geo sum: \(S=\dfrac{a_1}{1-r}\), \(|r|<1\)

Example

Find the 10th term of AP: \(3, 7, 11, \ldots\)

Answer: \(d=4\); \(a_{10}=3+9(4)=39\)

The Essential 20