Precalculus Exam 1 Study Guide: Rational Functions, Logarithms, Exponentials, and Trigonometry

Rational Functions

Domain of Rational Functions

The domain of a rational function consists of all real numbers except those that make the denominator zero. To find the domain, set the denominator equal to zero and solve for the excluded values.

• Key Point: 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 $Q(x) \neq 0$.

• Key Point: The domain is all real numbers except where $Q(x) = 0$.

• Example: For $H(x) = \frac{x^2 - 27}{x^2 - 5x + 6}$, set $x^2 - 5x + 6 = 0$ to find excluded values: $x = 2, 3$.

Graphing Rational Functions

To graph a rational function, follow these steps:

1. Find the domain by identifying values that make the denominator zero.

2. Find intercepts: Set numerator to zero for x-intercepts; set $x=0$ for y-intercept.

3. Identify vertical asymptotes (where denominator is zero and numerator is nonzero).

4. Identify horizontal or oblique asymptotes by comparing degrees of numerator and denominator.

5. Plot additional points as needed for accuracy.

• Example: $R(x) = \frac{x^2}{x^2 - x - 6}$ has vertical asymptotes at $x = 3, -2$ and a horizontal asymptote at $y = 1$.

Logarithmic and Exponential Equations

Solving Logarithmic Equations

Logarithmic equations can often be solved by using properties of logarithms and converting to exponential form.

• Key Point: $\log_b a = c$ is equivalent to $a = b^c$.

• Properties:

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

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

  • $\log_b (x^k) = k \log_b x$

• Example: $\log_2 (2x + 5) = 3$ implies $2x + 5 = 2^3 = 8$, so $x = 1.5$.

Solving Exponential Equations

To solve exponential equations, isolate the exponential term and take the logarithm of both sides if necessary.

• Key Point: $a^{f(x)} = b$ can be solved by taking $\log$ or $\ln$ of both sides.

• Example: $e^{5x} + 4 = 13$ leads to $e^{5x} = 9$, so $5x = \ln 9$, $x = \frac{\ln 9}{5}$.

Applications of Exponential and Logarithmic Functions

Compound Interest

Compound interest problems use the formula:

• For n times per year: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Compounded continuously: $A = Pe^{rt}$

• Key Terms: $A$ = final amount, $P$ = principal, $r$ = annual rate (decimal), $n$ = number of times compounded per year, $t$ = time in years.

• Example: $200$invested at $10\%$ compounded quarterly for $1.5$ years: $A = 200 \left(1 + \frac{0.10}{4}\right)^{4 \times 1.5}$

Present Value

To find the present value needed to reach a future amount:

• Formula: $P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$ or $P = Ae^{-rt}$ for continuous compounding.

• Example: To get $5000$after $1.75$ years at $8\%$ compounded daily: $P = \frac{5000}{\left(1 + \frac{0.08}{365}\right)^{365 \times 1.75}}$

Exponential Growth and Decay

Population growth and cooling problems use exponential models:

• Growth: $P(t) = P_0 e^{kt}$, where $k > 0$

• Decay: $P(t) = P_0 e^{-kt}$, where $k < 0$

• Doubling time: $k = \frac{\ln 2}{T_{double}}$

• Newton's Law of Cooling: $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$

• Example: If a population doubles every 21 months, $k = \frac{\ln 2}{21}$ (months), and $P(t) = 30000 e^{kt}$

Trigonometric Equations

Solving Basic Trigonometric Equations

To solve equations involving sine, cosine, or tangent, isolate the trigonometric function and use inverse functions.

• Key Point: $\sin \theta = a$ implies $\theta = \arcsin a + 2\pi n$ or $\pi - \arcsin a + 2\pi n$

• Key Point: $\cos \theta = a$ implies $\theta = \arccos a + 2\pi n$ or $-\arccos a + 2\pi n$

• Key Point: $\tan \theta = a$ implies $\theta = \arctan a + \pi n$

• Example: $4\sin \theta + 5 = 3$ leads to $\sin \theta = -0.5$, so $\theta = 7\pi/6, 11\pi/6$ (in $[0, 2\pi)$)

Solving Trigonometric Equations with Multiple Angles or Identities

Some equations require using trigonometric identities or factoring.

• Key Identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

• Example: $2\sin^2 \theta - \sin \theta - 1 = 0$ can be factored as $(2\sin \theta + 1)(\sin \theta - 1) = 0$

Solving Trigonometric Equations Involving Products and Sums

Equations may involve products or sums of trigonometric functions, requiring substitution or algebraic manipulation.

• Example: $\sin \theta (\cot \theta + \tan \theta) = \sec \theta$

• Rewrite $\cot \theta = \frac{\cos \theta}{\sin \theta}$ and $\tan \theta = \frac{\sin \theta}{\cos \theta}$, then simplify.

Summary Table: Key Formulas and Properties

Topic	Formula	Notes
Rational Function Domain	Set denominator $= 0$	Exclude these $x$ values
Compound Interest	$A = P \left(1 + \frac{r}{n}\right)^{nt}$	$n$ = times/year
Continuous Compounding	$A = Pe^{rt}$	Use $e$
Logarithm to Exponential	$\log_b a = c \iff a = b^c$	Change of form
Newton's Law of Cooling	$T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$	Temperature decay
Trigonometric Identity	$\sin^2 \theta + \cos^2 \theta = 1$	Pythagorean identity

Additional info: The study guide covers core Precalculus topics including rational functions, logarithmic and exponential equations, applications of exponential growth/decay, compound interest, and trigonometric equations. These are foundational for college-level Precalculus and are commonly tested on exams.