College Algebra Exam #3 Study Guide: Rational, Inverse, Exponential, and Logarithmic Functions

Section 4.5: Rational Functions

Understanding Rational Functions

Rational functions are a key topic in algebra, involving ratios of polynomials. Mastery of their properties and graphs is essential for solving complex equations and modeling real-world scenarios.

• 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.

• Asymptotes and Holes:

  • Vertical asymptotes occur at values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

  • Holes occur at values of $x$ where both $P(x)$ and $Q(x)$ are zero (common factors).

  • Horizontal asymptotes depend on 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 be an oblique/slant asymptote).

• Graphing: To graph a rational function:

  1. Find the domain.

  2. Identify and plot asymptotes and holes.

  3. Create a table of values to plot points.

  4. Sketch the curve, showing behavior near asymptotes.

• Example: For $f(x) = \frac{x^2 - 1}{x - 1}$, factor numerator: $f(x) = \frac{(x-1)(x+1)}{x-1}$. There is a hole at $x=1$ and the function simplifies to $f(x) = x+1$ for $x \neq 1$.

Section 5.1: Inverse Functions

Identifying and Graphing Inverse Functions

Inverse functions reverse the effect of the original function. Understanding their properties is crucial for solving equations and modeling inverse relationships.

• Definition: The inverse of a function $f$, denoted $f^{-1}$, satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f^{-1}$.

• One-to-One Functions: Only one-to-one functions (each $y$-value corresponds to exactly one $x$-value) have inverses that are also functions.

• Horizontal Line Test: A function has an inverse if and only if every horizontal line intersects its graph at most once.

• Finding the Inverse:

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

  2. Switch $x$ and $y$.

  3. Solve for $y$.

  4. Replace $y$ with $f^{-1}(x)$.

• Graphing: 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 $y = 2x + 3 \implies x = 2y + 3 \implies y = \frac{x-3}{2}$. Thus, $f^{-1}(x) = \frac{x-3}{2}$.

Section 5.2: Exponential Functions and Graphs

Properties and Graphs of Exponential Functions

Exponential functions model rapid growth or decay and are fundamental in many scientific and financial applications.

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

• Key Properties:

  • Domain: All real numbers.

  • Range: $(0, \infty)$ if $a > 0$.

  • Horizontal asymptote: $y = 0$.

  • Growth if $b > 1$; decay if $0 < b < 1$.

• Graphing: Exponential graphs pass through $(0, a)$ and increase or decrease rapidly.

• Example: $f(x) = 2^x$ is an increasing function with $y$-intercept at $(0,1)$.

Section 5.3: Logarithmic Functions and Graphs

Understanding and Graphing Logarithmic Functions

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

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

• Key Properties:

  • Domain: $(0, \infty)$

  • Range: All real numbers

  • Vertical asymptote: $x = 0$

  • Passes through $(1, 0)$

• Graphing: The graph increases slowly for large $x$ and is undefined for $x \leq 0$.

• Calculator Use: Use the LOG and LN keys for base 10 and base $e$ logarithms, respectively.

• Example: $f(x) = \log_{2}(x)$ passes through $(1,0)$ and $(2,1)$.

Section 5.4: Properties of Logarithmic Functions

Expanding and Combining Logarithmic Expressions

Logarithmic properties allow us to simplify, expand, or combine logarithmic expressions, which is essential for solving logarithmic equations.

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Power Rule: $\log_b(M^p) = p \cdot \log_b(M)$

• Example (Expanding): $\log_3(2x^4) = \log_3(2) + 4\log_3(x)$

• Example (Combining): $2\log_5(x) - \log_5(y) = \log_5\left(\frac{x^2}{y}\right)$

Section 5.5: Solving Exponential and Logarithmic Equations

Techniques for Solving Exponential and Logarithmic Equations

Solving these equations often requires applying properties of exponents and logarithms, and sometimes using calculators for approximate solutions.

• Solving Exponential Equations:

  • If possible, rewrite both sides with the same base and set exponents equal.

  • Otherwise, take logarithms of both sides and solve for the variable.

  • Example: $2^x = 7 \implies x = \log_2(7)$

• Solving Logarithmic Equations:

  • Combine logarithms into a single log if possible.

  • Rewrite the equation in exponential form to solve for the variable.

  • Example: $\log_3(x) = 4 \implies x = 3^4 = 81$

• Checking Solutions: Always check that solutions are in the domain of the original equation (e.g., arguments of logs must be positive).

• Approximate Solutions: Use calculators to find decimal approximations when exact answers are not possible.