College Algebra: Rational, Inverse, Exponential, and Logarithmic Functions Study Guide

Section 4.5 Rational Functions

Introduction to Rational Functions

Rational functions are a central topic in College Algebra, involving ratios of polynomials. Understanding their properties is essential for graphing and analyzing their behavior.

• Definition: A rational function is any function that can be written as $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Domain: The domain of a rational function consists of all real numbers except those that make the denominator zero.

• Vertical Asymptotes: Occur at values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$:

  • If degree of $P(x)$ < degree of $Q(x)$, horizontal asymptote at $y = 0$.

  • If degrees are equal, horizontal asymptote at $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$.

  • If degree of $P(x)$ > degree of $Q(x)$, no horizontal asymptote (may have an oblique/slant asymptote).

• Graphing: Use asymptotes, holes (removable discontinuities), intercepts, and a table of values to sketch the graph.

Example: For $f(x) = \frac{x^2 - 1}{x - 1}$, the domain is all real numbers except $x = 1$. There is a hole at $x = 1$ because both numerator and denominator are zero there.

Section 5.1 Inverse Functions

Understanding Inverse Functions

Inverse functions reverse the effect of the original function. They are crucial for solving equations and understanding function behavior.

• One-to-One Functions: A function is one-to-one if each output is produced by exactly one input. The horizontal line test is used to determine this.

• Finding Inverses: To find the inverse of $f(x)$, solve $y = f(x)$ for $x$ and then interchange $x$ and $y$.

• Verification: Two functions $f$ and $g$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their domains.

• Graphical Relationship: The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$.

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Section 5.2 Exponential Functions and Graphs

Introduction to Exponential Functions

Exponential functions model growth and decay processes. Their graphs have distinctive shapes and properties.

• Definition: An exponential function is of the form $f(x) = a \cdot b^x$, where $a \neq 0$, $b > 0$, and $b \neq 1$.

• Domain: All real numbers.

• Range: All positive real numbers if $a > 0$.

• Graph: Exponential growth if $b > 1$; exponential decay if $0 < b < 1$.

Example: $f(x) = 2^x$ is an exponential growth function.

Section 5.3 Logarithmic Functions and Graphs

Introduction to Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations involving exponentials.

• Definition: A logarithmic function is $f(x) = \log_b(x)$, where $b > 0$ and $b \neq 1$.

• Domain: $x > 0$.

• Range: All real numbers.

• Properties:

  • $\log_b(xy) = \log_b(x) + \log_b(y)$

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

  • $\log_b(x^r) = r \log_b(x)$

• Change-of-Base Formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ for any base $k$.

Example: $\log_{10}(100) = 2$ because $10^2 = 100$.

Section 5.4 Properties of Logarithms

Expanding and Condensing Logarithmic Expressions

Logarithmic properties allow us to expand or condense expressions for simplification and solving equations.

• Expanding: Use product, quotient, and power rules to write logarithms as sums and differences.

• Condensing: Combine multiple logarithms into a single logarithm using the same rules in reverse.

Example: $\log_2(8x^2) = \log_2(8) + \log_2(x^2) = 3 + 2\log_2(x)$

Section 5.5 Solving Exponential and Logarithmic Equations

Techniques for Solving Equations

Solving exponential and logarithmic equations is essential for applications in science, finance, and engineering.

• Exponential Equations: Set both sides to the same base and equate exponents, or use logarithms to solve for the variable.

• Logarithmic Equations: Combine logarithms and rewrite in exponential form to solve for the variable.

• Checking Solutions: Always check proposed solutions in the original equation to avoid extraneous answers.

• Exact and Approximate Solutions: Some equations require calculator use for decimal approximations.

Example: Solve $2^x = 10$. Take logarithms: $x = \log_2(10) \approx 3.32$.

Summary Table: Properties of Exponential and Logarithmic Functions

Function Type	General Form	Domain	Range	Key Properties
Exponential	$f(x) = a \cdot b^x$	All real numbers	$a > 0$: $(0, \infty)$	Growth/decay, horizontal asymptote at $y=0$
Logarithmic	$f(x) = \log_b(x)$	$x > 0$	All real numbers	Inverse of exponential, vertical asymptote at $x=0$