College Algebra Test 3 Review – Step-by-Step Guidance

Q1. Given $f(x) = \frac{3}{x+2}$, find the following:

• Domain

• Range

• Horizontal asymptote

• y-intercept

• Graph

Background

Topic: Rational Functions

This question tests your understanding of rational functions, including how to determine their domain, range, asymptotes, intercepts, and how to sketch their graphs.

Key Terms and Formulas

• Domain: The set of all real numbers for which the function is defined.

• Range: The set of all possible output values (y-values).

• Horizontal Asymptote: A horizontal line that the graph approaches as $x$ goes to infinity or negative infinity.

• y-intercept: The point where the graph crosses the y-axis ($x=0$).

• General form of a rational function: $f(x) = \frac{a}{x+b}$

Step-by-Step Guidance

1. Domain: Set the denominator not equal to zero and solve for $x$:

   $x + 2 \neq 0$

   Solve for $x$ to find the values that are excluded from the domain.

2. Range: Consider the possible values of $f(x)$ as $x$ varies over its domain. Think about what values $f(x)$ cannot take, especially considering the horizontal asymptote.

3. Horizontal Asymptote: For $f(x) = \frac{3}{x+2}$, as $x$ approaches infinity or negative infinity, what value does $f(x)$ approach?

4. y-intercept: Set $x = 0$ and solve for $f(0)$:

   $f(0) = \frac{3}{0+2}$

5. Graph: Use the information above to sketch the graph. Mark the asymptote, intercept, and note the behavior near the excluded value of $x$.

   

Try solving on your own before revealing the answer!

Final Answers:

• Domain: $x \neq -2$

• Range: $y \neq 0$

• Horizontal Asymptote: $y = 0$

• y-intercept: $(0, 1.5)$

• Graph: The graph is a hyperbola with a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = 0$.

Each part follows from the properties of rational functions. The domain excludes the value that makes the denominator zero, and the horizontal asymptote is determined by the degrees of the numerator and denominator.

Q2. Write $32 = 8^x$ in logarithmic form.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to convert between exponential and logarithmic forms.

Key Terms and Formulas

• Exponential form: $a = b^x$

• Logarithmic form: $x = \log_b a$

Step-by-Step Guidance

1. Identify the base ($b$), exponent ($x$), and result ($a$) in the equation $32 = 8^x$.

2. Rewrite the equation in logarithmic form using the definition: $x = \log_b a$.

Try solving on your own before revealing the answer!

Final Answer:

$x = \log_8 32$

This expresses the exponent $x$ as a logarithm with base 8 and argument 32.