College Algebra: Functions and Domain Study Guide

Q1. Find the domain of each function:

• CO2: f(x) = x^2 + 12

• CO2: f(x) = 2x - 3

• CO2: f(x) = x^2 - 3x + 5

• CO2: f(x) = \frac{2}{x-4}

• CO2: f(x) = \sqrt{x-4}

Background

Topic: Domain of Functions

This question tests your understanding of how to determine the set of all possible input values (domain) for different types of functions, including polynomials, rational functions, and radical functions.

Key Terms and Formulas

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

• Polynomial functions: Defined for all real numbers.

• Rational functions: Undefined where the denominator is zero.

• Radical functions (even root): Defined where the radicand is non-negative.

Step-by-Step Guidance

1. For each function, identify its type (polynomial, rational, or radical).

2. For polynomial functions (like $f(x) = x^2 + 12$), recall that the domain is usually all real numbers.

3. For rational functions (like $f(x) = \frac{2}{x-4}$), set the denominator not equal to zero and solve for $x$.

4. For radical functions with an even root (like $f(x) = \sqrt{x-4}$), set the radicand greater than or equal to zero and solve for $x$.

5. Write the domain in interval notation for each function.

Try solving on your own before revealing the answer!

Final Answer:

• $f(x) = x^2 + 12$: Domain is $(-\infty, \infty)$

• $f(x) = 2x - 3$: Domain is $(-\infty, \infty)$

• $f(x) = x^2 - 3x + 5$: Domain is $(-\infty, \infty)$

• $f(x) = \frac{2}{x-4}$: Domain is $(-\infty, 4) \cup (4, \infty)$

• $f(x) = \sqrt{x-4}$: Domain is $[4, \infty)$

Polynomials are defined everywhere, rational functions are undefined where the denominator is zero, and even root radical functions are defined where the radicand is non-negative.

Q2. Find the domain of the following functions:

• CO2: $f(x) = \frac{1}{x^2 - 9}$

• CO2: $f(x) = \sqrt{x+2}$

• CO2: $f(x) = \frac{1}{\sqrt{x-1}}$

Background

Topic: Domain of Rational and Radical Functions

This question focuses on finding the domain for functions that involve denominators and square roots, which have specific restrictions.

Key Terms and Formulas

• Rational function: Undefined where the denominator is zero.

• Square root function: Defined where the radicand is greater than or equal to zero.

• Square root in denominator: The radicand must be strictly greater than zero (since denominator cannot be zero).

Step-by-Step Guidance

1. For $f(x) = \frac{1}{x^2 - 9}$, set $x^2 - 9 \neq 0$ and solve for $x$.

2. For $f(x) = \sqrt{x+2}$, set $x+2 \geq 0$ and solve for $x$.

3. For $f(x) = \frac{1}{\sqrt{x-1}}$, set $x-1 > 0$ and solve for $x$.

4. Express each domain in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

• $f(x) = \frac{1}{x^2 - 9}$: Domain is $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$

• $f(x) = \sqrt{x+2}$: Domain is $[-2, \infty)$

• $f(x) = \frac{1}{\sqrt{x-1}}$: Domain is $(1, \infty)$

Remember, denominators cannot be zero and square roots require non-negative radicands (or positive if in the denominator).