Rational, Exponential, and Logarithmic Functions: Study Guide for Exam 4 (Math 225)

Rational Functions

Domain, Holes, and Vertical Asymptotes

Rational functions are functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0. Understanding their domain and discontinuities is essential for graphing and analysis.

• Domain: The domain consists of all real numbers except those that make the denominator zero. To find excluded values, solve q(x) = 0.

• Holes (Removable Discontinuities): Occur when a factor cancels from both numerator and denominator. The function is undefined at this x-value, but the graph has a 'hole' rather than an asymptote.

• Vertical Asymptotes (VA): Occur at zeros of the denominator that remain after simplification. The function approaches infinity near these x-values.

Example: For $f(x) = \frac{x^2-4}{x^2-x-6}$, factor to get $\frac{(x-2)(x+2)}{(x-3)(x+2)}$. There is a hole at $x = -2$ (common factor), and a vertical asymptote at $x = 3$.

Horizontal and Slant Asymptotes

Asymptotes describe the end behavior of rational functions. Horizontal and slant (oblique) asymptotes depend on the degrees of the numerator and denominator.

• Horizontal Asymptote (HA): If degrees are equal, HA is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$. If degree of denominator > numerator, HA is $y = 0$.

• Slant Asymptote (SA): If degree of numerator is one more than denominator, perform polynomial division. The quotient (without remainder) is the slant asymptote.

Example: For $f(x) = \frac{x^2+1}{x-1}$, divide to get $x + 1 + \frac{2}{x-1}$, so SA is $y = x + 1$.

Intercepts and Graphing

Intercepts are points where the graph crosses the axes. Combine all features to sketch rational functions.

• x-intercepts: Set numerator equal to zero (excluding holes).

• y-intercept: Evaluate $f(0)$ if defined.

• Graphing: Use domain, holes, asymptotes, intercepts, and behavior near asymptotes to sketch the function.

Example: For $f(x) = \frac{x^2-4}{x-1}$, x-intercepts at $x = \pm2$, y-intercept at $(0, 4)$, VA at $x = 1$, and SA from division.

Exponential and Logarithmic Functions

Exponential Functions and Their Graphs

Exponential functions have the form f(x) = a \cdot b^x, where a ≠ 0 and b > 0, b ≠ 1. They model growth and decay processes.

• Growth: If $b > 1$, the function increases as x increases.

• Decay: If $0 < b < 1$, the function decreases as x increases.

• y-intercept: At $x = 0$, $f(0) = a$.

• Horizontal Asymptote: Typically $y = 0$.

Example: $f(x) = 3 \cdot 2^x$ is growth (b = 2 > 1). $g(x) = 5 \cdot (\frac{1}{3})^x$ is decay, y-intercept 5, HA $y = 0$.

Compound and Continuous Interest

Exponential functions are used in financial applications such as compound and continuous interest.

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

• Continuous Interest: $A = P e^{rt}$

• Variables: P = principal, r = annual rate, n = number of compounding periods per year, t = time in years.

Example: $100 at 5\%$ compounded continuously for 2 years: $A = 100e^{0.05 \times 2}$.

Example: $500$doubling at $4\%$ compounded monthly: $1000 = 500 \left(1 + \frac{0.04}{12}\right)^{12t}$.

Logarithmic Functions: Form, Evaluation, and Graphs

Logarithmic functions are the inverses of exponential functions. The general form is y = \log_b(x), where b > 0, b ≠ 1.

• Inverse Relationship: $y = \log_b(x)$ means $b^y = x$.

• Common Logarithm: $\log(x)$ is base 10.

• Natural Logarithm: $\ln(x)$ is base $e$.

• Domain: Argument must be strictly positive ($x > 0$).

• Vertical Asymptote: At $x = 0$.

Example: $\log_2(16) = y$ means $2^y = 16 \Rightarrow y = 4$.

Example: Rewrite $\ln(7) = x$ as $e^x = 7$.

Example: Domain of $f(x) = \log(x-5)$: $x-5 > 0 \Rightarrow x > 5$.

Example: Domain of $g(x) = \ln(x^2-9)$: $x^2-9 > 0 \Rightarrow x < -3$ or $x > 3$.

Logarithm Rules, Equations, and Inequalities

Properties of Logarithms

Logarithms follow several important properties that allow expansion and condensation of expressions.

• 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: $\log_b(M^k) = k \log_b(M)$

Example: Expand $\log_3(9x)$: $\log_3(9) + \log_3(x) = 2 + \log_3(x)$.

Example: Condense $2 \ln(x) - \frac{1}{2} \ln(y) + 3 \ln(z)$: $\ln\left(\frac{x^2 z^3}{\sqrt{y}}\right)$.

Change of Base Formula

The change of base formula allows rewriting logarithms in terms of common or natural logs.

• Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ (where c is any valid base, often 10 or e)

Example: $\log_5(12) = \frac{\ln(12)}{\ln(5)}$

Example: $\log_7(x) = \frac{\log(x)}{\log(7)}$

Solving Exponential Equations

Exponential equations involve variables in exponents. They can be solved by rewriting with common bases or by taking logarithms.

• Rewrite with common base: If possible, express both sides with the same base and equate exponents.

• Take logarithms: If bases differ, apply logarithms to both sides and solve for the variable.

Example: $3^x = 27 \Rightarrow 3^x = 3^3 \Rightarrow x = 3$

Example: $5^{2x-1} = 7 \Rightarrow \ln(5^{2x-1}) = \ln(7) \Rightarrow (2x-1)\ln(5) = \ln(7) \Rightarrow x = \frac{\ln(7)}{2\ln(5)} + \frac{1}{2}$

Solving Logarithmic Equations

Logarithmic equations require isolating the logarithm, converting to exponential form, and checking for extraneous solutions.

• Isolate the logarithm: Move all log terms to one side.

• Convert to exponential form: $\log_b(x) = y \Rightarrow x = b^y$

• Check domain: Ensure solutions satisfy log argument restrictions.

Example: $\log_2(x-3) = 4 \Rightarrow x-3 = 2^4 = 16 \Rightarrow x = 19$

Example: $\log_2(x) + \log_2(x-2) = 3 \Rightarrow \log_2(x(x-2)) = 3 \Rightarrow x^2-2x = 8 \Rightarrow x^2-2x-8=0$; solutions $x=4, -2$. Discard $x=-2$ (domain restriction), so $x=4$.

Solving Exponential and Logarithmic Inequalities

Solving inequalities involves algebraic manipulation and understanding the monotonicity of exponential and logarithmic functions.

• Exponential Inequalities: Rewrite in terms of common base or take logarithms, then solve.

• Logarithmic Inequalities: Isolate log, convert to exponential, and check domain.

• Express solution set: Use interval notation.

Example: $2^x > 8 \Rightarrow 2^x > 2^3 \Rightarrow x > 3$

Example: $\ln(x-1) \leq 0 \Rightarrow x-1 \leq 1 \Rightarrow 1 < x \leq 2$

Summary Table: Logarithm Properties

Property	Formula	Example
Product Rule	$\log_b(MN) = \log_b(M) + \log_b(N)$	$\log_3(9x) = \log_3(9) + \log_3(x)$
Quotient Rule	$\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$	$\log_2\left(\frac{x}{y}\right) = \log_2(x) - \log_2(y)$
Power Rule	$\log_b(M^k) = k \log_b(M)$	$2 \ln(x) = \ln(x^2)$
Change of Base	$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$	$\log_5(12) = \frac{\ln(12)}{\ln(5)}$

Additional info: All examples and formulas are expanded for clarity and completeness. This guide covers all core skills for Exam 4 as outlined in the provided materials.