Precalculus Final Exam Practice: Functions, Exponentials, Logarithms, and Trigonometry

Practice Problems for Precalculus Final Exam

Overview

This set of practice problems covers key Precalculus topics, including functions and their domains, exponential and logarithmic equations, and trigonometric functions and identities. The problems are designed to reinforce conceptual understanding and problem-solving skills in preparation for a cumulative final exam.

Functions and Their Graphs

Domain and Range of Functions

• Domain: The set of all possible input values (x-values) for which a function is defined.

• Range: The set of all possible output values (y-values) a function can produce.

• To find the domain, look for restrictions such as division by zero or even roots of negative numbers.

Example: For $f(x) = \sqrt{x-2}$, the domain is $x \geq 2$ because the expression under the square root must be non-negative.

Composite Functions

• Composite function: $(f \circ g)(x) = f(g(x))$.

• To find the domain of $f(g(x))$, the output of $g(x)$ must be in the domain of $f$.

Example: If $f(x) = \sqrt{x}$ and $g(x) = x-1$, then $f(g(x)) = \sqrt{x-1}$, so the domain is $x \geq 1$.

Inverse Functions

• Inverse function: $f^{-1}(x)$ undoes the action of $f(x)$.

• To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Not all functions have inverses; the function must be one-to-one (pass the horizontal line test).

Example: For $f(x) = 2x + 3$, solve $y = 2x + 3$ for $x$: $x = \frac{y-3}{2}$. Thus, $f^{-1}(x) = \frac{x-3}{2}$.

Exponential and Logarithmic Functions

Exponential Functions

• Exponential function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Domain: $(-\infty, \infty)$; Range: $(0, \infty)$ for $a > 0$.

• Exponential growth and decay are modeled by $y = y_0 e^{kt}$, where $k > 0$ for growth and $k < 0$ for decay.

Example: $y = 3e^{2x}$ is an exponential growth function.

Logarithmic Functions

• Logarithmic function: $f(x) = \log_a x$, the inverse of $a^x$.

• Domain: $(0, \infty)$; Range: $(-\infty, \infty)$.

• Properties:

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

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

  • $\log_a (x^r) = r \log_a x$

Example: $\log_2 8 = 3$ because $2^3 = 8$.

Solving Exponential and Logarithmic Equations

• To solve $a^x = b$, take the logarithm of both sides: $x = \log_a b$.

• To solve $\log_a x = b$, rewrite as $x = a^b$.

Example: Solve $2^x = 16$. $x = \log_2 16 = 4$.

Trigonometric Functions

Basic Trigonometric Functions

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Other functions: cosecant, secant, cotangent

Example: For a right triangle with sides 3, 4, 5, $\sin \theta = 3/5$ if the opposite side is 3 and hypotenuse is 5.

Unit Circle and Exact Values

• The unit circle allows for finding exact values of trigonometric functions at special angles (e.g., $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$).

• Coordinates $(x, y)$ on the unit circle correspond to $(\cos \theta, \sin \theta)$.

Example: $\sin \frac{\pi}{6} = \frac{1}{2}$, $\cos \frac{\pi}{3} = \frac{1}{2}$.

Trigonometric Identities

• Pythagorean identities:

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

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

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

• Sum and difference formulas:

  • $\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

  • $\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

  • $\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

• Double-angle formulas:

  • $\sin 2\theta = 2 \sin \theta \cos \theta$

  • $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

• Half-angle formulas:

  • $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

  • $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

Example: $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$.

Solving Trigonometric Equations

• Isolate the trigonometric function, then use inverse functions or identities to solve for the variable.

• Consider all solutions in the given interval, often $[0, 2\pi)$ or $[0, 360^\circ)$.

Example: Solve $2\sin x = 1$ for $x$ in $[0, 2\pi)$: $\sin x = \frac{1}{2}$, so $x = \frac{\pi}{6}, \frac{5\pi}{6}$.

Applications of Exponential, Logarithmic, and Trigonometric Functions

Exponential Growth and Decay

• Population growth, radioactive decay, and compound interest are modeled by $A(t) = A_0 e^{kt}$.

• To solve for time or rate, use logarithms to isolate the variable.

Example: If $A_0 = 100$, $A(t) = 50$ when $t = 10$, solve $50 = 100e^{10k}$ for $k$.

Logarithmic Applications

• Logarithms are used to solve for time in exponential models, such as half-life problems.

• Change of base formula: $\log_b a = \frac{\log_c a}{\log_c b}$ (commonly $c = 10$ or $e$).

Example: To solve $2^x = 7$, take $\log$ of both sides: $x = \frac{\log 7}{\log 2}$.

Trigonometric Applications

• Modeling periodic phenomena: $y = A \sin(Bx + C) + D$

• Amplitude: $|A|$; Period: $\frac{2\pi}{|B|}$; Phase shift: $-\frac{C}{B}$

• Applications include sound waves, tides, and circular motion.

Example: For $y = 3\sin(2x - \frac{\pi}{2})$, amplitude is 3, period is $\pi$, phase shift is $\frac{\pi}{4}$.

Solving Triangles and Law of Sines/Cosines

Law of Sines

• $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Used when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Example: Given $A = 30^\circ$, $B = 45^\circ$, $a = 10$, find $b$.

Law of Cosines

• $c^2 = a^2 + b^2 - 2ab\cos C$

• Used when given two sides and the included angle (SAS) or all three sides (SSS).

Example: Given $a = 5$, $b = 7$, $C = 60^\circ$, find $c$.

Summary Table: Key Properties of Exponential and Logarithmic Functions

Function	Domain	Range	Key Properties
$f(x) = a^x$	$(-\infty, \infty)$	$(0, \infty)$	Exponential growth/decay; $a > 0$, $a \neq 1$
$f(x) = \log_a x$	$(0, \infty)$	$(-\infty, \infty)$	Inverse of $a^x$; $a > 0$, $a \neq 1$

Summary Table: Key Trigonometric Values

Angle	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0$	$0$	$1$	$0$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$\frac{\pi}{2}$	$1$	$0$	Undefined

Additional info:

• This study guide is based on a comprehensive set of practice problems for a Precalculus final exam, covering chapters on functions, exponentials, logarithms, and trigonometry, including analytic and applied aspects.

• Students should be familiar with algebraic manipulation, properties of logarithms and exponents, and the use of trigonometric identities for simplification and equation solving.

• Applications include modeling with exponential and trigonometric functions, solving triangles, and interpreting real-world scenarios using mathematical models.