Precalculus Final Exam Review – Comprehensive Study Notes

Functions and Their Domains

Finding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Square Root Functions: The expression under the square root must be non-negative.

• Rational Functions: The denominator cannot be zero.

• Logarithmic Functions: The argument of the logarithm must be positive.

Example:

• For $f(x) = \sqrt{x-3}$, domain: $x \geq 3$

• For $g(x) = \frac{1}{x^2-9}$, domain: $x \neq 3, x \neq -3$

Asymptotes and Graphing Rational Functions

Horizontal and Vertical Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches.

• Vertical Asymptotes: Occur where the denominator of a rational function is zero (and the numerator is not zero at that point).

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator polynomials.

Example:

• For $f(x) = \frac{2x}{x-1}$, vertical asymptote at $x=1$, horizontal asymptote at $y=2$.

Graphing Functions

Key Steps in Graphing

• Identify intercepts (where the graph crosses the axes).

• Plot at least three points for accuracy.

• Label asymptotes and important features.

Example:

• For $f(x) = x^2 - 3x + 1$, plot points for $x = -1, 0, 1, 2, 3$ and sketch the parabola.

Polynomial and Rational Functions

Finding Zeros of Polynomials

The zeros of a polynomial are the x-values where the function equals zero. These can be found by factoring or using the quadratic formula.

• For higher-degree polynomials, use factoring, synthetic division, or the Rational Root Theorem.

Example:

• For $f(x) = x^2 - 5x + 6$, zeros are $x=2$ and $x=3$.

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions for different intervals of the domain.

• Graph each piece on its specified interval.

• Check for open or closed circles at endpoints depending on the inclusion of the endpoint.

Example:

• $f(x) = \begin{cases} x^2, & x < 1 \\ 2x+1, & x \geq 1 \end{cases}$

Absolute and Local Extrema

Identifying Maximum and Minimum Values

Absolute maximum/minimum is the highest/lowest point on the entire graph. Local maximum/minimum is the highest/lowest point in a small neighborhood.

• Use the graph to identify these points visually.

Trigonometric Functions and Graphs

Graphing and Analyzing Trigonometric Functions

• State the amplitude, period, phase shift, and vertical shift.

• Domain and range depend on the function (e.g., sine and cosine have domain $(-\infty, \infty)$ and range $[-1, 1]$).

Example:

• $y = 2\sin(x-\frac{\pi}{2})$ has amplitude 2, period $2\pi$, phase shift $\frac{\pi}{2}$ right.

Polar and Parametric Equations

Graphing and Converting

• Polar equations use $r$ and $\theta$; parametric equations use $x(t)$ and $y(t)$.

• Convert between forms using $x = r\cos\theta$, $y = r\sin\theta$.

Logarithmic and Exponential Functions

Solving Logarithmic Equations

• Use properties of logarithms to combine or expand expressions.

• Remember: $\log_b(xy) = \log_b x + \log_b y$ and $\log_b(x^k) = k\log_b x$.

Example:

• Solve $\log_2(x) + \log_2(x-3) = 2$ by combining logs and solving the resulting quadratic.

Trigonometric Identities and Equations

Verifying and Solving

• Use fundamental identities: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, etc.

• Solve equations by isolating the trigonometric function and using the unit circle.

Example:

• Verify $\sec y \tan y = \frac{\sin y}{\cos^2 y}$ by rewriting in terms of sine and cosine.

Inverse Functions

Finding and Verifying Inverses

• To find the inverse, solve for $x$ in terms of $y$, then swap $x$ and $y$.

• A function has an inverse if it is one-to-one (passes the horizontal line test).

Example:

• For $f(x) = 2x+1$, inverse is $f^{-1}(x) = \frac{x-1}{2}$.

Exponents and Simplification

Properties of Exponents

• $a^m \cdot a^n = a^{m+n}$

• $(a^m)^n = a^{mn}$

• $\frac{a^m}{a^n} = a^{m-n}$

Example:

• Simplify $\frac{a^7}{a^3} = a^{4}$

Difference Quotient

Definition

The difference quotient is used to find the average rate of change of a function:

$\frac{f(x+h) - f(x)}{h}$

Example:

• For $f(x) = x^2$, $\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2 - x^2}{h} = 2x + h$

Coordinate Systems

Converting Between Polar and Rectangular Coordinates

• Rectangular to polar: $r = \sqrt{x^2 + y^2}$, $\theta = \arctan(\frac{y}{x})$

• Polar to rectangular: $x = r\cos\theta$, $y = r\sin\theta$

Summary Table: Key Function Properties

Function Type	Domain	Range	Asymptotes
Polynomial	$(-\infty, \infty)$	$(-\infty, \infty)$	None
Rational	All real except where denominator = 0	Varies	Vertical & Horizontal
Exponential	$(-\infty, \infty)$	$(0, \infty)$	Horizontal
Logarithmic	$(0, \infty)$	$(-\infty, \infty)$	Vertical
Trigonometric	Varies	Varies	Vertical (tan, sec, csc, cot)

Additional info: These notes synthesize the main topics and question types from the review sheet, providing definitions, examples, and key properties for Precalculus exam preparation.