College Algebra Exam 2 Review: Step-by-Step Guidance

Q1. Determine the implicit domain of $f(x) = \frac{3x-1}{7x+4}$.

Background

Topic: Domains of Rational Functions

This question tests your ability to find the domain of a rational function, which is the set of all real numbers for which the function is defined.

Key Terms and Formulas:

• Rational function: A function of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.

• Domain: The set of all real numbers $x$ for which the function is defined (i.e., the denominator is not zero).

Step-by-Step Guidance

1. Identify the denominator: $7x + 4$.

2. Set the denominator not equal to zero: $7x + 4 \neq 0$.

3. Solve for $x$ to find the value(s) that make the denominator zero.

4. State the domain as all real numbers except the value(s) found in the previous step. Express your answer in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

The denominator is zero when $7x + 4 = 0 \implies x = -\frac{4}{7}$.

So, the domain is all real numbers except $x = -\frac{4}{7}$.

Interval notation: $(-\infty, -\frac{4}{7}) \cup (-\frac{4}{7}, \infty)$

Q2. Determine the implicit domain of $g(x) = \frac{-2x+6}{x^2 + 3x - 10}$.

Background

Topic: Domains of Rational Functions

This question asks you to find the domain of a rational function by identifying values that make the denominator zero.

Key Terms and Formulas:

• Denominator: The function is undefined when the denominator equals zero.

• Factoring: You may need to factor the denominator to find all values that make it zero.

Step-by-Step Guidance

1. Identify the denominator: $x^2 + 3x - 10$.

2. Set the denominator equal to zero: $x^2 + 3x - 10 = 0$.

3. Factor the quadratic equation to find the values of $x$ that make the denominator zero.

4. Exclude these $x$-values from the domain. Express the domain in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

Factoring: $x^2 + 3x - 10 = (x + 5)(x - 2)$, so the denominator is zero at $x = -5$ and $x = 2$.

Domain: $(-\infty, -5) \cup (-5, 2) \cup (2, \infty)$

Q3. Determine the implicit domain of $h(x) = \sqrt{-7x + 11}$.

Background

Topic: Domains of Radical Functions

This question tests your ability to find the domain of a function involving a square root, which is defined only when the radicand (expression under the root) is non-negative.

Key Terms and Formulas:

• Radicand: The expression inside the square root must be greater than or equal to zero for real-valued functions.

Step-by-Step Guidance

1. Set the radicand greater than or equal to zero: $-7x + 11 \geq 0$.

2. Solve the inequality for $x$.

3. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$-7x + 11 \geq 0 \implies x \leq \frac{11}{7}$

Domain: $(-\infty, \frac{11}{7}]$

Q4. Determine the implicit domain of $s(x) = \frac{x^2 - 1}{\sqrt{-3 + 5x}}$.

Background

Topic: Domains of Rational and Radical Functions

This question combines both rational and radical expressions. The denominator must be nonzero and the radicand must be positive (since it's in the denominator).

Key Terms and Formulas:

• Radicand in denominator: The expression under the square root must be strictly greater than zero (not just non-negative) to avoid division by zero.

Step-by-Step Guidance

1. Set the radicand greater than zero: $-3 + 5x > 0$.

2. Solve the inequality for $x$.

3. Express the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$-3 + 5x > 0 \implies x > \frac{3}{5}$

Domain: $(\frac{3}{5}, \infty)$

Q5. Let $f(x) = 4 + \frac{2}{x}$ and $g(x) = \sqrt{5x}$. Find $(f+g)(x)$, $(g-f)(x)$, $f \cdot g(x)$, and $\frac{g}{f}(x)$. Give the domain for each in both set-builder and interval notation.

Background

Topic: Operations with Functions and Domains

This question tests your ability to perform addition, subtraction, multiplication, and division of functions, and to determine the domain of the resulting functions.

Key Terms and Formulas:

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(g-f)(x) = g(x) - f(x)$

• Product: $f \cdot g(x) = f(x) \cdot g(x)$

• Quotient: $\frac{g}{f}(x) = \frac{g(x)}{f(x)}$ (where $f(x) \neq 0$)

• Domain: The intersection of the domains of $f$ and $g$, and for the quotient, exclude where $f(x) = 0$.

Step-by-Step Guidance

1. Find the domain of $f(x)$: $x \neq 0$ (since division by zero is undefined).

2. Find the domain of $g(x)$: $5x \geq 0 \implies x \geq 0$ (since the radicand must be non-negative).

3. For each operation, the domain is the intersection of the domains of $f$ and $g$.

4. For the quotient $\frac{g}{f}(x)$, further exclude values where $f(x) = 0$.

5. Write each combined function and state the domain in set-builder and interval notation, but stop before the final answer.

Try solving on your own before revealing the answer!

Final Answer:

Domains:

• For $f(x)$: $x \neq 0$

• For $g(x)$: $x \geq 0$

• Intersection: $x > 0$

• For $\frac{g}{f}(x)$, $f(x) = 0$ when $4 + \frac{2}{x} = 0 \implies x = -\frac{1}{2}$, but this is not in the intersection domain ($x > 0$), so $x > 0$ for all operations.

Set-builder notation: $\{x \mid x > 0\}$

Interval notation: $(0, \infty)$

Each combined function is:

• $(f+g)(x) = 4 + \frac{2}{x} + \sqrt{5x}$

• $(g-f)(x) = \sqrt{5x} - \left(4 + \frac{2}{x}\right)$

• $f \cdot g(x) = \left(4 + \frac{2}{x}\right) \cdot \sqrt{5x}$

• $\frac{g}{f}(x) = \frac{\sqrt{5x}}{4 + \frac{2}{x}}$

Q6. Using the formulas from the previous question, determine $(f+g)(2)$, $(g-f)(5)$, $f \cdot g(20)$, and $\frac{g}{f}(0)$.

Background

Topic: Evaluating Functions

This question asks you to substitute specific values into the combined functions you found previously.

Key Terms and Formulas:

• Recall the formulas for $f(x)$ and $g(x)$ from the previous question.

• Plug in the given $x$-values into each function and simplify.

Step-by-Step Guidance

1. For each part, substitute the given $x$-value into the appropriate formula.

2. For example, to find $(f+g)(2)$, compute $f(2)$ and $g(2)$, then add them.

3. Repeat for each requested value, but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer:

• $(f+g)(2) = 4 + \frac{2}{2} + \sqrt{5 \cdot 2} = 4 + 1 + \sqrt{10} \approx 7.16$

• $(g-f)(5) = \sqrt{5 \cdot 5} - (4 + \frac{2}{5}) = \sqrt{25} - (4 + 0.4) = 5 - 4.4 = 0.6$

• $f \cdot g(20) = (4 + \frac{2}{20}) \cdot \sqrt{5 \cdot 20} = (4 + 0.1) \cdot \sqrt{100} = 4.1 \cdot 10 = 41$

• $\frac{g}{f}(0)$ is undefined because $f(0)$ is undefined (division by zero).