Precalculus Study Guide: Functions, Graphs, Domains, and Equations

Functions and Their Graphs

Solving Inequalities

Solving inequalities is a fundamental skill in precalculus, often requiring interval notation to express solution sets.

• Key Point 1: To solve an inequality, isolate the variable and determine the intervals where the inequality holds.

• Key Point 2: Express solutions using interval notation, which clearly indicates the range of values satisfying the inequality.

• Example: Solve $2x + 2 eq 6$.

  • Subtract 2: $2x eq 4$

  • Divide by 2: $x eq 2$

  • Interval notation: $(-\infty, 2) \\cup (2, \\infty)$

Functions: Domains and Operations

Finding Domains

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

• Key Point 1: For rational functions, exclude values that make the denominator zero.

• Key Point 2: For functions involving square roots, ensure the radicand is non-negative.

• Example: Find the domain of $f(x) = rac{2}{x-1}$.

  • Denominator cannot be zero: $x-1 eq 0 \Rightarrow x eq 1$

  • Domain: $(-\\infty, 1) \\cup (1, \\infty)$

Function Operations

Functions can be added, subtracted, multiplied, or divided, and their domains are determined by the intersection of the domains of the individual functions and any new restrictions.

• Key Point 1: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$).

• Example: If $f(x) = x^2$ and $g(x) = x-2$, then $(f+g)(x) = x^2 + x - 2$.

Composite Functions

A composite function $f(g(x))$ applies one function to the result of another. The domain of the composite is restricted by both functions.

• Key Point 1: The domain of $f(g(x))$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• Example: If $f(x) = \sqrt{x}$ and $g(x) = x-2$, then $f(g(x)) = \sqrt{x-2}$, domain: $x-2 \geq 0 \Rightarrow x \geq 2$.

Inverse Functions

Finding and Verifying Inverses

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(x)$ maps $y$ back to $x$.

• Key Point 1: To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Key Point 2: The domain of $f$ becomes the range of $f^{-1}$, and vice versa.

• Example: For $f(x) = x + 2$, $y = x + 2 \Rightarrow x = y - 2$, so $f^{-1}(x) = x - 2$.

Graphing Functions

Intercepts and Asymptotes

Understanding the graphical features of functions is essential for analysis and interpretation.

• Key Point 1: x-intercept: Set $y=0$ and solve for $x$.

• Key Point 2: y-intercept: Set $x=0$ and solve for $y$.

• Key Point 3: Vertical asymptotes occur where the denominator of a rational function is zero (and not canceled).

• Key Point 4: Horizontal asymptotes are determined by the degrees of the numerator and denominator.

• Example: For $f(x) = \frac{2x}{x^2 - 4}$:

  • Vertical asymptotes: $x^2 - 4 = 0 \Rightarrow x = \pm 2$

  • Horizontal asymptote: Degree numerator (1) < degree denominator (2), so $y=0$

Exponential and Logarithmic Functions

Properties and Equations

Exponential and logarithmic functions are inverses of each other and are used to model growth and decay.

• Key Point 1: $y = a^x$ is exponential; $y = \log_a x$ is logarithmic.

• Key Point 2: To solve $\log_a x = b$, rewrite as $x = a^b$.

• Example: Solve $\log_3 x = 4$.

  • $x = 3^4 = 81$

Solving Equations

Quadratic and Rational Equations

Solving equations is a core skill, including quadratic, rational, and logarithmic forms.

• Key Point 1: Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Key Point 2: Rational equations require finding a common denominator and checking for extraneous solutions.

• Example: Solve $x^2 - 5x + 4 = 0$.

  • Factor: $(x-4)(x-1) = 0 \Rightarrow x = 4, 1$

Summary Table: Types of Equations and Solution Methods

Type	General Form	Solution Method	Example
Linear	$ax + b = 0$	Isolate $x$	$2x + 3 = 7 \Rightarrow x = 2$
Quadratic	$ax^2 + bx + c = 0$	Factoring, quadratic formula	$x^2 - 5x + 4 = 0$
Rational	$\frac{P(x)}{Q(x)} = 0$	Set numerator to zero, check denominator	$\frac{2}{x-1} = 0$
Logarithmic	$\log_a x = b$	Rewrite as exponential	$\log_3 x = 4 \Rightarrow x = 81$

*Additional info: Some explanations and examples have been expanded for clarity and completeness based on standard precalculus curriculum topics.*