Q1. Find the domain of each rational function:

Background

Topic: Rational Functions and Domain

This question tests your understanding of how 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 where and are polynomials.

• Domain: All real values of for which .

Step-by-Step Guidance

1. Identify the denominator: .

2. Set the denominator equal to zero: .

3. Solve for to find values that make the denominator zero.

4. Exclude these values from the domain.

Try solving on your own before revealing the answer!

Q2. Write each rational expression in lowest terms:

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor and reduce rational expressions to their simplest form.

Key Terms and Formulas:

• Factorization: Expressing a polynomial as a product of its factors.

• Lowest terms: A rational expression is in lowest terms when the numerator and denominator have no common factors.

Step-by-Step Guidance

1. Factor the numerator: can be written as .

2. Factor the denominator: can be written as .

3. Identify any common factors in the numerator and denominator.

4. Cancel the common factors to simplify the expression.

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Q3. Perform the indicated operation. If possible, simplify your answer:

Background

Topic: Adding Rational Expressions

This question tests your ability to add rational expressions with common denominators.

Key Terms and Formulas:

• Common denominator: When denominators are the same, add numerators directly.

Step-by-Step Guidance

1. Check if the denominators are the same (they are: ).

2. Add the numerators: .

3. Write the result over the common denominator.

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Q4. Simplify each complex fraction:

Background

Topic: Complex Fractions

This question tests your ability to simplify fractions where the numerator and denominator are themselves fractions.

Key Terms and Formulas:

• Complex fraction: A fraction in which the numerator or denominator (or both) contain fractions.

• To simplify:

Step-by-Step Guidance

1. Rewrite the complex fraction as a division: .

2. Recall that dividing by a fraction is the same as multiplying by its reciprocal.

3. Multiply: .

4. Simplify the expression by canceling common factors.

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Q5. Solve each equation for :

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations involving rational expressions.

Key Terms and Formulas:

• Rational equation: An equation containing rational expressions.

• To solve: Isolate by multiplying both sides by .

Step-by-Step Guidance

1. Multiply both sides by to eliminate the denominator.

2. Rewrite the equation: .

3. Isolate by dividing both sides by 4.

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Q6. The product of one more than a number and twice the reciprocal of the number is . Find the number.

Background

Topic: Rational Equations and Word Problems

This question tests your ability to translate a word problem into a rational equation and solve for the unknown.

Key Terms and Formulas:

• Reciprocal: For a number , the reciprocal is .

• Product: Multiplication of two expressions.

Step-by-Step Guidance

1. Let the unknown number be .

2. One more than the number: .

3. Twice the reciprocal: .

4. Set up the equation: .

5. Multiply out and solve for (stop before the final calculation).

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Q7. 400 feet hits the ground at a speed of 160 feet per second. Find the height of a cliff over the gorge if a dropped ant hits the ground at a speed of 128 feet per second.

Background

Topic: Applications of Rational Equations (Physics Context)

This question tests your ability to use rational equations in a real-world context, specifically relating speed and height in free fall.

Key Terms and Formulas:

• Free fall speed: , where is final speed, is acceleration due to gravity, is height.

• Proportional relationships: If and are speeds from heights and , .

Step-by-Step Guidance

1. Write the formula for speed in free fall: .

2. Set up a proportion using the two speeds and heights: .

3. Solve for by squaring both sides and isolating .

Try solving on your own before revealing the answer!