Intermediate Algebra Study Guide: Rational Expressions and Functions

Q1. Find the domain of each rational function.

Background

Topic: Rational Functions and Domain

This question tests your understanding of how to determine 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 $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.

• Domain: All real values of $x$ except those that make the denominator zero.

Step-by-Step Guidance

1. Identify the denominator of the rational function.

2. Set the denominator equal to zero: $Q(x) = 0$.

3. Solve the equation for $x$ to find values that are not allowed in the domain.

4. Exclude these values from the domain.

Try solving on your own before revealing the answer!

Final Answer:

The domain is all real numbers except those values of $x$ that make the denominator zero. For each function, solve $Q(x) = 0$ and exclude those solutions from the domain.

Q2. Simplify the following rational expression.

Background

Topic: Simplifying Rational Expressions

This question tests your ability to reduce rational expressions to their lowest terms by factoring and canceling common factors.

Key Terms and Formulas:

• Factor: Write the numerator and denominator as products of their factors.

• Simplify: Cancel common factors from the numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator and denominator completely.

2. Identify any common factors between the numerator and denominator.

3. Cancel the common factors.

4. Write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer:

The rational expression is simplified by factoring and canceling common factors. The result is the lowest terms form.

Q3. Simplify the following rational expression.

Background

Topic: Simplifying Rational Expressions

This question is similar to Q2 and tests your ability to factor and reduce rational expressions.

Key Terms and Formulas:

• Factorization: Express polynomials as products of their factors.

• Lowest Terms: The simplest form of the rational expression.

Step-by-Step Guidance

1. Factor both the numerator and denominator.

2. Look for common factors.

3. Cancel the common factors.

4. Express the result in lowest terms.

Try solving on your own before revealing the answer!

Final Answer:

The simplified form is obtained by factoring and canceling common factors.

Q4. Simplify: $\frac{2x^2 - 11x - 21}{x^2 - 8x + 7}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor quadratic expressions and simplify rational expressions.

Key Terms and Formulas:

• Quadratic Factorization: Factor expressions of the form $ax^2 + bx + c$.

• Cancel Common Factors: Remove factors that appear in both numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator $2x^2 - 11x - 21$.

2. Factor the denominator $x^2 - 8x + 7$.

3. Identify any common factors between numerator and denominator.

4. Cancel the common factors and write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{2x^2 - 11x - 21}{x^2 - 8x + 7} = \frac{(2x + 3)(x - 7)}{(x - 7)(x - 1)} = \frac{2x + 3}{x - 1}$

The factor $(x - 7)$ cancels from both numerator and denominator.

Q5. Divide and simplify: $\frac{a + b}{ab} \div \frac{a^2 - b^2}{2a^3b}$

Background

Topic: Dividing Rational Expressions

This question tests your ability to divide rational expressions and simplify the result.

Key Terms and Formulas:

• Division of Rational Expressions: $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Rewrite the division as multiplication by the reciprocal.

2. Factor $a^2 - b^2$ as $(a + b)(a - b)$.

3. Multiply the numerators and denominators.

4. Cancel any common factors.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{a + b}{ab} \div \frac{a^2 - b^2}{2a^3b} = \frac{a + b}{ab} \times \frac{2a^3b}{a^2 - b^2} = \frac{2a^3b(a + b)}{ab(a^2 - b^2)}$

After factoring and canceling, the simplified result is $\frac{2a^2}{a - b}$.

Q6. Find the area of the rectangle with sides $\frac{x + 9}{x}$ m and $\frac{2x}{x^2 - 81}$ m.

Background

Topic: Multiplying Rational Expressions (Area)

This question tests your ability to multiply rational expressions to find the area of a rectangle.

Key Terms and Formulas:

• Area of Rectangle: $\text{Area} = \text{length} \times \text{width}$

• Factoring: $x^2 - 81 = (x + 9)(x - 9)$

Step-by-Step Guidance

1. Write the area as the product of the two rational expressions.

2. Factor $x^2 - 81$ in the denominator.

3. Multiply the numerators and denominators.

4. Cancel any common factors.

Try solving on your own before revealing the answer!

Final Answer:

Area $= \frac{x + 9}{x} \times \frac{2x}{x^2 - 81} = \frac{2(x + 9)x}{x(x + 9)(x - 9)}$

After canceling $(x + 9)$ and $x$, the area simplifies to $\frac{2}{x - 9}$ m2.