College Algebra: Sample Exam 1 Study Guide

Functions and Their Domains

Definition and Evaluation of Functions

In College Algebra, a function is a relation that assigns each input exactly one output. The domain of a function is the set of all possible input values (typically x-values) for which the function is defined.

• Evaluating Functions: Substitute the given value into the function and simplify.

• Domain Restrictions: Functions may be undefined for certain values, such as division by zero or taking the square root of a negative number.

Examples:

• For $f(x) = \frac{1}{x^2 - 4x - 5}$, the domain excludes values of $x$ that make the denominator zero.

• For $f(x) = \sqrt{12 - 2x}$, the domain includes values of $x$ such that $12 - 2x \geq 0$.

Piecewise Functions: These are defined by different expressions depending on the input value.

• Example: $h(x) = \begin{cases} -3x - 18 & x < -5 \\ 1 & -5 \leq x < 1 \\ x + 2 & x \geq 1 \end{cases}$

Linear Equations and Slope

Finding the Slope Between Two Points

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x between two points.

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: For points $\left(-\frac{3}{4}, -\frac{1}{4}\right)$ and $\left(\frac{5}{7}, -\frac{5}{7}\right)$, substitute into the formula to find the slope.

Slope and y-Intercept from Equation

Linear equations can be written in the form $ax + by = c$ or $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• To find the slope and y-intercept: Rearrange the equation to $y = mx + b$ form.

• Example: For $3x - 4y = 20$, solve for $y$ to get $y = \frac{3}{4}x - 5$; slope is $\frac{3}{4}$, y-intercept is $-5$.

Graphing Linear Equations

To graph a line, use the slope and y-intercept:

• Plot the y-intercept on the y-axis.

• Use the slope to find another point.

Solving Linear Equations

Single and Multi-Step Equations

Solving linear equations involves isolating the variable using algebraic operations.

• Example: $3x - 5 = 2x + 1$

• Subtract $2x$ from both sides: $x - 5 = 1$

• Add $5$to both sides:$x = 6$

Function Composition

Definition and Domain

Function composition involves applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• To find the domain of the composition, consider the domains of both functions and ensure the output of $g(x)$ is in the domain of $f(x)$.

• Example: If $f(x) = x^5 - 2$ and $g(x) = \sqrt{x + 2}$, $(f \circ g)(x) = f(g(x)) = (\sqrt{x + 2})^5 - 2$.

Word Problems and Applications

Taxi Fare Problem

Word problems often require setting up equations based on the scenario.

• Example: Taxi fare is $4.30 plus $2.40 per mile. If the total fare is $30.90, set up the equation: $4.30 + 2.40m = 30.90$ and solve for $m$.

Perimeter Problem

Perimeter problems involve geometric formulas.

• Example: For a rectangular farm with perimeter 45 meters, let length = $l$, width = $w$.

• Perimeter formula: $2l + 2w = 45$

• Solve for one variable in terms of the other.

Paycheck and Overtime

Calculate regular and overtime pay using given rates.

• Example: If Deisy earns time and a half (1.5 times regular wage) for hours over 40, set up equations to find her regular hourly wage.

Piecewise Functions

Evaluating Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the input.

• Identify which interval the input value falls into, then use the corresponding expression.

• Example: For $h(x)$ as defined above, $h(-5)$ uses the second case, $h(0)$ uses the second case, $h(1)$ uses the third case, $h(4)$ uses the third case.

Summary Table: Types of Questions

Topic	Example Question	Key Concept
Function Domain	Find the domain of $f(x) = \sqrt{12 - 2x}$	Domain restrictions
Slope Between Points	Find the slope between $(-\frac{3}{4}, -\frac{1}{4})$ and $(\frac{5}{7}, -\frac{5}{7})$	Slope formula
Linear Equation	Find slope and y-intercept of $3x - 4y = 20$	Standard and slope-intercept forms
Function Composition	Find $(f \circ g)(x)$ and domain	Composition and domain
Word Problem	Taxi fare calculation	Equation setup and solution
Piecewise Function	Evaluate $h(-5), h(0), h(1), h(4)$	Piecewise evaluation

Additional info: Some questions reference specific textbook sections (e.g., Section 1.2, 1.3, etc.), which may provide further examples and explanations. The study guide above covers all major topics presented in the sample exam and is suitable for College Algebra exam preparation.