Domain and Range of a function and its Inverse function
5. The domain of function f(x) is (−∞, 6) and its range is (−5, ∞).

Question at position 4
4
 
1 point 

 
 
Question at position 4 
a. What is the domain of f−1(x)?


a. What is the domain of f−1(x)?


(−∞, 6)


(−5, ∞)


(−6, ∞)


(−∞, 5)

Question at position 5
5
 
1 point 

 
 
Question at position 5 
b. What is the range of f−1(x)?


b. What is the range of f−1(x)?


(−∞, 6)


(−6, ∞)


(−∞, 5)


(−5, ∞)

For a function \( f(x) \) and its inverse \( f^{-1}(x) \), the **domain** of the inverse function is the same as the **range** of the original function, and the **range** of the inverse function is the same as the **domain** of the original function.

Given the following information for the original function \( f(x) \):
- The domain of \( f(x) \) is \( (-\infty, 6) \)
- The range of \( f(x) \) is \( (-5, \infty) \)

### Part a: Domain of \( f^{-1}(x) \)
The **domain of the inverse function** is the **range** of the original function, which is \( (-5, \infty) \).

So the correct answer is:
- **Domain of \( f^{-1}(x) \):** \( (-5, \infty) \)

### Part b: Range of \( f^{-1}(x) \)
The **range of the inverse function** is the **domain** of the original function, which is \( (-\infty, 6) \).

So the correct answer is:
- **Range of \( f^{-1}(x) \):** \( (-\infty, 6) \)

### Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How is the inverse of a function found algebraically?
2. What are the graphical relationships between a function and its inverse?
3. Can all functions have inverses? Why or why not?
4. What is the domain and range of a composite function \( f(f^{-1}(x)) \)?
5. How do you determine if a function is one-to-one and therefore has an inverse?

**Tip:** The graph of a function and its inverse is symmetric along the line \( y = x \).

The domain of function f(x) is (−∞, 6) and its range is (−5, ∞). What is the domain and range of f−1(x)?

Learn how to find the domain and range of an inverse function based on the original function f(x). In this problem, the domain of f(x) is (−∞, 6) and the range is (−5, ∞), which helps determine the domain and range of f−1(x). Suitable for Grades 10-12.

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