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

Functions and Their Domains

Understanding Domains

The domain of a function is the set of all possible input values (typically x-values) for which the function is defined. Determining the domain often involves identifying values that would cause division by zero, taking the square root of a negative number, or other undefined operations.

• Rational Functions: The domain excludes values that make the denominator zero.

• Square Root Functions: The domain includes values that make the radicand (expression under the root) non-negative.

• Composite Functions: The domain is restricted by both the inner and outer functions.

Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$.

Example: For $f(x) = \sqrt{x-2}$, the domain is $x \geq 2$.

Solving Inequalities

Interval Notation

When solving inequalities, the solution is often expressed in interval notation, which describes the set of values that satisfy the inequality.

• Example: $x^2 \geq 6x$ → $x \leq 0$ or $x \geq 6$ → $(-\infty, 0] \cup [6, \infty)$

• Example: $2x^2 - 5x < 2$ → $x \in (-\infty, -0.4) \cup (2.5, \infty)$

Function Operations and Composition

Sum, Difference, Product, and Quotient of Functions

Given two functions $f(x)$ and $g(x)$, you can create new functions by adding, subtracting, multiplying, or dividing them:

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

Composition of Functions

The composition of functions $f$ and $g$ is written as $f(g(x))$, meaning you substitute $g(x)$ into $f(x)$.

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

Inverse Functions

Finding the Inverse

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find the inverse:

1. Replace $f(x)$ with $y$.

2. Swap $x$ and $y$.

3. Solve for $y$.

Example: For $f(x) = x + 2$, the inverse is $f^{-1}(x) = x - 2$.

Graphing Functions

Intercepts and Asymptotes

When graphing rational functions, it is important to identify:

• x-intercepts: Where $f(x) = 0$.

• y-intercepts: Where $x = 0$.

• Vertical asymptotes: Values of $x$ that make the denominator zero (and are not canceled by the numerator).

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

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

• Vertical asymptotes at $x = 2$ and $x = -2$.

• Horizontal asymptote at $y = 0$.

• x-intercept at $x = 0$.

Exponential and Logarithmic Equations

Solving Logarithmic Equations

To solve equations involving logarithms, use properties such as:

• $\log_a(xy) = \log_a x + \log_a y$

• $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• $\log_a(x^k) = k \log_a x$

Example: Solve $\log_3(x^2 - 3x) = 10$:

• Rewrite as $x^2 - 3x = 3^{10}$

• Solve the quadratic equation for $x$.

Solving Systems of Equations

Linear and Quadratic Systems

Systems of equations can be solved by substitution, elimination, or graphing. For quadratic systems, set equations equal and solve for the variables.

• Example: $x^2 - 5x - 16 = 0$ → Solve for $x$ using the quadratic formula:

$x = \frac{5 \pm \sqrt{25 + 64}}{2} = \frac{5 \pm \sqrt{89}}{2}$

Summary Table: Function Properties

Function Type	Domain	Key Features
Rational	All real numbers except where denominator = 0	Vertical/horizontal asymptotes, intercepts
Square Root	Radicand $\geq 0$	Endpoint, increasing/decreasing behavior
Logarithmic	Argument $> 0$	Vertical asymptote, domain restrictions

Additional info:

• Some questions involve graphing and identifying intercepts and asymptotes, which are key skills in Precalculus.

• Interval notation is used throughout to express solution sets.

• Inverse functions and composition are foundational for understanding function behavior.