StudyBlaze

Building a Foundation

1. What is the notation used to represent the inverse of a function \( f(x) \)? (1 choice)

C) \( \frac{1}{f(x)} \)

D) \( f'(x) \)

A) \( f^2(x) \)

B) \( f^{-1}(x) \)

2. Which of the following statements are true about inverse functions? (multiple choice)

B) A function must be one-to-one to have an inverse.

D) The graph of an inverse function is a reflection over the line \( y = x \).

C) The inverse of a function is always a function.

A) The inverse of a function reverses the operation of the original function.

3. Explain the Horizontal Line Test and its significance in determining if a function has an inverse.

4. List two conditions necessary for a function to have an inverse.

Condition 1

Condition 2

Your answers will be evaluated by AI for key concepts, not exact wording. Focus on main ideas.

5. If a function \( f(x) \) is increasing over its entire domain, what can be said about its inverse? (1 choice)

D) The inverse is a constant function.

C) The inverse is decreasing.

A) The inverse does not exist.

B) The inverse is also increasing.

Application and Analysis

6. Given \( f(x) = 2x + 3 \), what is \( f^{-1}(x) \)? (1 choice)

A) \( \frac{x - 3}{2} \)

B) \( 2x - 3 \)

C) \( \frac{x + 3}{2} \)

D) \( 2(x - 3) \)

7. Which of the following functions have inverses? (multiple choice)

A) \( f(x) = x^2 \) over \( x \geq 0 \)

B) \( f(x) = \sin(x) \) over \( -\frac{\pi}{2} \) \( \text{to} \) \( \frac{\pi}{2} \)

C) \( f(x) = e^x \)

D) \( f(x) = x^3 \)

8. Solve for the inverse of the function \( f(x) = \frac{1}{x+1} \).

9. If \( f(x) = \sqrt{x} \) and \( g(x) = x^2 \), are these functions inverses of each other? (1 choice)

D) No, because \( f(g(x)) \neq x \) for all \( x \).

A) Yes, because \( f(g(x)) = x \) for all \( x \).

C) Yes, because both functions are one-to-one.

B) No, because \( g(f(x)) \neq x \) for all \( x \).

10. Discuss the impact of restricting the domain of a function on its inverse. Provide an example.

Evaluation and Creation

11. Which of the following statements best evaluates the necessity of inverse functions in real-world applications? (1 choice)

B) Inverse functions are essential for solving equations and converting units.

C) Inverse functions are only useful in theoretical mathematics.

D) Inverse functions complicate mathematical models unnecessarily.

A) Inverse functions are rarely used in practical scenarios.

12. Evaluate the following scenarios and identify where inverse functions are applicable: (multiple choice)

B) Calculating the original price from a discounted price

D) Solving for the principal amount in compound interest

A) Converting Celsius to Fahrenheit

C) Determining the time taken given speed and distance

13. Create a real-world problem that involves finding the inverse of a function. Explain how you would solve it and the significance of the inverse in your scenario.

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