Study Guide: Rational, Exponential, and Logarithmic Functions (Chapters 3 & 4)

Rational Functions and Their Properties

Definition and Domain

A rational function is any function that can be written as the ratio of two polynomials, i.e., $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$. The domain of a rational function consists of all real numbers except those for which the denominator is zero.

• Key Point: To determine the domain, set the denominator equal to zero and solve for excluded values.

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

Graphs and Asymptotes

Rational functions often have vertical asymptotes at values excluded from the domain and may have horizontal or oblique asymptotes depending on the degrees of the numerator and denominator.

• Vertical Asymptotes: Occur at zeros of the denominator.

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

• Oblique Asymptotes: Occur when the degree of the numerator is one higher than the denominator.

• Example: $f(x) = \frac{x^2}{x-1}$ has a vertical asymptote at $x=1$ and an oblique asymptote.

Intercepts and Applications

• x-intercepts: Set numerator equal to zero.

• y-intercepts: Evaluate $f(0)$.

• Applications: Rational functions model rates, concentrations, and other real-world ratios.

Exponential Functions

Definition and Properties

An exponential function has the form $f(x) = a \cdot b^x$, where $a \neq 0$, $b > 0$, and $b \neq 1$. Exponential functions model rapid growth or decay.

• Key Point: The base $b$ determines growth ($b > 1$) or decay ($0 < b < 1$).

• Example: $f(x) = 2^x$ is exponential growth; $f(x) = (0.5)^x$ is exponential decay.

Graphing and Intercepts

• Graph: Exponential functions are always positive and increase or decrease rapidly.

• x-intercept: May not exist for standard exponential functions.

• y-intercept: $f(0) = a$.

Solving Exponential Equations

• Exact Solutions: Use properties of exponents or logarithms.

• Decimal Approximations: Estimate solutions to desired accuracy.

• Example: Solve $2^x = 8$; $x = 3$.

Applications

• Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$

• Uninhibited Growth/Decay: $N(t) = N_0 e^{kt}$

• Example: Population growth, radioactive decay.

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function: $f(x) = \log_b(x)$, where $b > 0$, $b \neq 1$. Logarithms answer the question: "To what exponent must $b$ be raised to get $x$?"

• Key Point: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$.

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

Graphing Logarithmic Functions

• Domain: $x > 0$

• Range: All real numbers

• Vertical Asymptote: At $x = 0$

Logarithmic Equations and Properties

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^r) = r \cdot \log_b(x)$

• Change of Base: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$

Solving Logarithmic Equations

• Isolate the logarithm and rewrite in exponential form.

• Example: Solve $\log_2(x) = 3$; $x = 8$.

Applications

• Modeling: Logarithmic functions are used in pH, sound intensity, and population models.

• Financial Applications: Used in compound interest and exponential decay.

Summary Table: Exponential vs. Logarithmic Functions

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

Additional info:

• These notes are based on exam objectives for Chapters 3 and 4, covering rational, exponential, and logarithmic functions, their properties, graphs, and applications.

• Images included are directly relevant to the exam objectives and reinforce the study content.