The function f left parenthesis x right parenthesis equals 2 x plus 7 is​ one-to-one.

a. Find an equation for f Superscript negative 1​, the inverse function.

b. Verify that your equation is correct by showing that f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisequalsx and f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisequalsx.

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Part 1

a. Select the correct choice below and fill in the answer​ box(es) to complete your choice.

​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)

A.

f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xnot equals
  
enter your response here

B.

f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xless than or equals
  
enter your response here

C.

f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xgreater than or equals
  
enter your response here

D.

f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for all x

### Part 1a: Finding the Inverse Function

The function given is \( f(x) = 2x + 7 \). To find the inverse function, follow these steps:

1. Replace \( f(x) \) with \( y \):
   \[
   y = 2x + 7
   \]

2. Swap \( x \) and \( y \) (this is the key step to finding the inverse):
   \[
   x = 2y + 7
   \]

3. Solve for \( y \):
   \[
   x - 7 = 2y
   \]
   \[
   y = \frac{x - 7}{2}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x - 7}{2}
\]

### Part 1b: Verifying the Inverse Function

To verify that \( f^{-1}(x) = \frac{x - 7}{2} \) is correct, we need to check two conditions:

1. **\( f(f^{-1}(x)) = x \):**

   Substitute \( f^{-1}(x) \) into \( f(x) \):
   \[
   f(f^{-1}(x)) = f\left( \frac{x - 7}{2} \right)
   \]
   Using the original function \( f(x) = 2x + 7 \):
   \[
   f\left( \frac{x - 7}{2} \right) = 2\left( \frac{x - 7}{2} \right) + 7 = (x - 7) + 7 = x
   \]
   So, \( f(f^{-1}(x)) = x \).

2. **\( f^{-1}(f(x)) = x \):**

   Substitute \( f(x) = 2x + 7 \) into \( f^{-1}(x) \):
   \[
   f^{-1}(f(x)) = f^{-1}(2x + 7) = \frac{(2x + 7) - 7}{2} = \frac{2x}{2} = x
   \]
   So, \( f^{-1}(f(x)) = x \).

Both conditions are satisfied, confirming that \( f^{-1}(x) = \frac{x - 7}{2} \) is the correct inverse function.

### Answer Choice:
The correct answer is:
\[
\boxed{f^{-1}(x) = \frac{x - 7}{2}, \text{ for all } x.}
\]

---

Would you like further clarification or details on any step?

### 5 Related Questions:
1. What are the general steps to finding an inverse of a linear function?
2. How can you verify if a function is one-to-one before finding its inverse?
3. Why is it necessary to swap \( x \) and \( y \) when finding an inverse function?
4. Can a function have an inverse if it is not one-to-one? Why or why not?
5. How would the process change if the function were not linear?

### Tip:
To ensure a function is one-to-one, check that it passes the horizontal line test, meaning no horizontal line intersects the graph more than once.

The function f(x) = 2x + 7 is one-to-one. a. Find an equation for f^(-1), the inverse function. b. Verify that your equation is correct by showing that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Learn how to find the inverse of the linear function f(x) = 2x + 7 and verify it by using key algebraic concepts. The solution includes step-by-step methods for finding the inverse and checking the properties of inverse functions. Suitable for students in Grades 10-12.

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