Q1. Find an equation of the inverse relation for .

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a linear function by switching the roles of and and solving for $y$.

Key Terms and Formulas:

• Inverse function: If is a function, its inverse undoes the action of $f(x)$.

• To find the inverse, swap and and solve for $y$.

Step-by-Step Guidance

1. Start with the original equation: .

2. Switch and to get .

3. Solve this new equation for by isolating $y$ on one side.

4. Once you have by itself, write the inverse relation as

Try solving on your own before revealing the answer!

Q2. Find an equation of the inverse relation for .

Background

Topic: Inverse Relations (Nonlinear)

This question asks you to find the inverse of a relation involving both and in a nonlinear way.

Key Terms and Formulas:

• Inverse relation: Swap and and solve for $y$.

• Be careful with exponents and algebraic manipulation.

Step-by-Step Guidance

1. Start with the given equation: .

2. Switch and to get .

3. Solve for in terms of by isolating $y$.

4. Express the inverse relation as

Try solving on your own before revealing the answer!

Q3. Graph as a solid line. Graph its inverse as a dashed line by reflecting across .

Background

Topic: Graphing Functions and Their Inverses

This question tests your understanding of how to graph a function and its inverse, and how the inverse is a reflection across the line .

Key Terms and Formulas:

• Inverse graph: The graph of the inverse is a reflection of the original graph across the line .

• To graph the inverse, swap the and coordinates of points on the original graph.

Step-by-Step Guidance

1. Plot the graph of as a solid curve.

2. Draw the line as a reference (usually a dashed diagonal line).

3. To graph the inverse, reflect each point on the original graph to .

4. Sketch the reflected (inverse) curve as a dashed line.

Try sketching the graphs before revealing the answer!

Q4. Find the inverse of the function .

Background

Topic: Inverse Functions (Linear)

This question asks you to find the inverse of a simple linear function.

Key Terms and Formulas:

• Inverse function: undoes the action of .

• To find the inverse, swap and and solve for $y$.

Step-by-Step Guidance

1. Write .

2. Swap and to get .

3. Solve for by dividing both sides by 7.

4. Write the inverse function as

Try solving on your own before revealing the answer!

Q5. Find the inverse of the function .

Background

Topic: Inverse Functions (Linear with Fractional Coefficient)

This question tests your ability to find the inverse of a linear function with a fractional coefficient.

Key Terms and Formulas:

• Inverse function: Swap and and solve for $y$.

• Be careful with fractions when isolating .

Step-by-Step Guidance

1. Write .

2. Swap and to get .

3. Subtract 6 from both sides to isolate the term with .

4. Multiply both sides by 4 to solve for .

5. Write the inverse function as

Try solving on your own before revealing the answer!

Q6. For and , use composition to show that is the inverse of .

Background

Topic: Verifying Inverses Using Composition

This question asks you to use function composition to verify that two functions are inverses.

Key Terms and Formulas:

• Composition: and

• If and are true inverses, both compositions should return .

Step-by-Step Guidance

1. Compute by substituting into .

2. Simplify the expression to see if it equals .

3. Compute by substituting into .

4. Simplify this expression as well.

5. If both simplify to , the functions are inverses.

Try working through the compositions before revealing the answer!

Q7. For and , use composition to show that is the inverse of .

Background

Topic: Verifying Inverses Using Composition (Rational Functions)

This question asks you to verify that two rational functions are inverses by using composition.

Key Terms and Formulas:

• Composition: and

• If both compositions return , the functions are inverses.

Step-by-Step Guidance

1. Compute by plugging into .

2. Simplify the resulting expression step by step.

3. Compute by plugging into .

4. Simplify this expression as well.

5. Check if both compositions simplify to .

Try working through the algebra before revealing the answer!