College Algebra Study Guide: Functions, Equations, Graphs, and Transformations

Q1. Solve the equation: $5x - 2 = 7$

Background

Topic: Linear Equations

This question tests your ability to solve a basic linear equation for the variable $x$.

Key Terms and Formulas:

• Linear equation: An equation of the form $ax + b = c$.

• Solving for $x$: Isolate $x$ on one side of the equation.

Step-by-Step Guidance

1. Start by adding $2$ to both sides to eliminate the constant term on the left.

2. Divide both sides by $5$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 1.8$

$5x - 2 = 7 \Rightarrow 5x = 9 \Rightarrow x = \frac{9}{5} = 1.8$

We isolated $x$ and solved using basic algebraic steps.

Q2. Solve and graph the solution in interval notation: $3x - 2 + 4 \geq 20$

Background

Topic: Linear Inequalities

This question tests your ability to solve a linear inequality and express the solution using interval notation.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear expression.

• Interval notation: A way to represent the set of solutions.

Step-by-Step Guidance

1. Combine like terms on the left side: $-2 + 4$.

2. Add or subtract terms to isolate $3x$.

3. Divide both sides by $3$to solve for$x$.

4. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer: $x \geq 6$

$3x + 2 \geq 20 \Rightarrow 3x \geq 18 \Rightarrow x \geq 6$

Interval notation: $[6, \infty)$

Q3. Solve and graph the solution in interval notation: $2x - 1 + 7 < 13$

Background

Topic: Linear Inequalities

This question tests your ability to solve a linear inequality and express the solution using interval notation.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear expression.

• Interval notation: A way to represent the set of solutions.

Step-by-Step Guidance

1. Combine like terms: $-1 + 7$.

2. Isolate $2x$ on one side.

3. Divide both sides by $2$to solve for$x$.

4. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer: $x < 3.5$

$2x + 6 < 13 \Rightarrow 2x < 7 \Rightarrow x < 3.5$

Interval notation: $(-\infty, 3.5)$

Q4. Determine the domain of $f(x) = x + 3$

Background

Topic: Domain of Functions

This question tests your understanding of the domain of a linear function.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Recognize that $f(x) = x + 3$ is a linear function.

2. Linear functions are defined for all real numbers.

Try solving on your own before revealing the answer!

Final Answer: Domain is $(-\infty, \infty)$

Linear functions have no restrictions on their domain.

Q5. Determine the domain of $f(x) = x + 5x^2 + x - 42$

Background

Topic: Domain of Polynomial Functions

This question tests your understanding of the domain of a polynomial function.

Key Terms and Formulas:

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

• Polynomial function: An expression involving powers of $x$.

Step-by-Step Guidance

1. Identify the function as a polynomial.

2. Polynomial functions are defined for all real numbers.

Try solving on your own before revealing the answer!

Final Answer: Domain is $(-\infty, \infty)$

Polynomial functions have no restrictions on their domain.

Q6. Determine the domain of $f(x) = x + 5x - 7$

Background

Topic: Domain of Linear Functions

This question tests your understanding of the domain of a linear function.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Recognize that $f(x) = x + 5x - 7$ is a linear function.

2. Linear functions are defined for all real numbers.

Try solving on your own before revealing the answer!

Final Answer: Domain is $(-\infty, \infty)$

Linear functions have no restrictions on their domain.

Q7. For the following graph of $f(x)$, find:

• Domain

• Range

• Interval(s) where increasing

• $f(0)$

• The value(s) of $x$ for which $f(x) = -1$

Background

Topic: Graph Analysis

This question tests your ability to interpret a graph and identify key features such as domain, range, intervals of increase, and specific function values.

Key Terms and Formulas:

• Domain: All $x$-values for which the graph exists.

• Range: All $y$-values the graph attains.

• Increasing interval: Where the graph moves upward as $x$ increases.

• $f(0)$: The $y$-value when $x = 0$.

• $f(x) = -1$: Find $x$ where the graph crosses $y = -1$.

Step-by-Step Guidance

1. Examine the graph to determine the leftmost and rightmost $x$-values for the domain.

2. Identify the lowest and highest $y$-values for the range.

3. Look for intervals where the graph is rising as you move from left to right.

4. Find the $y$-value at $x = 0$ by locating the point where the graph crosses the $y$-axis.

5. Locate the $x$-value(s) where the graph crosses $y = -1$.

Try solving on your own before revealing the answer!

Final Answer:

• Domain: $[-4, \infty)$

• Range: $[-1, \infty)$

• Increasing: $(-2, \infty)$

• $f(0) = 1$

• $f(x) = -1$ when $x = -2$

These values are determined by analyzing the graph's endpoints, turning points, and intersections.