Question

Complete parts​ (a) and​ (b) below.
a. Write the inverse of ​f(x)equals2xminus3.
b. Do the values for f Superscript negative 1 in the provided table fit the equation for f Superscript negative 1 Baseline question mark
x
f left parenthesis x right parenthesis
x
f Superscript negative 1 Baseline left parenthesis x right parenthesis
negative 1
negative 5
negative 5
negative 1
0
negative 3
negative 3
0
1
negative 1
negative 1
1
2
1
1
2
3
3
3
3
Question content area bottom
Part 1
a. The inverse of​ f(x) is f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here.
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)
Part 2
b. Do the values for f Superscript negative 1 in the provided table fit the equation for f Superscript negative 1​?

Answer 1

To solve the problem, we'll start with part (a) and then move to part (b).

### Part (a)
To find the inverse of the function \( f(x) = 2x - 3 \), we follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = 2x - 3
\]

2. **Swap \( x \) and \( y \)**:
\[
x = 2y - 3
\]

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

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x + 3}{2}
\]

### Part (b)
Next, we will check if the values for \( f^{-1} \) in the provided table fit the equation \( f^{-1}(x) = \frac{x + 3}{2} \).

#### Provided Table:
| \( x \) | \( f(x) \) | \( f^{-1}(x) \) |
|---------|------------|------------------|
| -1 | -5 | -1 |
| 0 | -3 | 0 |
| 1 | -1 | 1 |
| 2 | 1 | 2 |
| 3 | 3 | 3 |

Now we need to calculate \( f^{-1}(x) \) for each of these \( x \) values using our derived inverse function \( f^{-1}(x) = \frac{x + 3}{2} \).

1. For \( x = -1 \):
\[
f^{-1}(-1) = \frac{-1 + 3}{2} = \frac{2}{2} = 1
\]
2. For \( x = 0 \):
\[
f^{-1}(0) = \frac{0 + 3}{2} = \frac{3}{2} = 1.5
\]
3. For \( x = 1 \):
\[
f^{-1}(1) = \frac{1 + 3}{2} = \frac{4}{2} = 2
\]
4. For \( x = 2 \):
\[
f^{-1}(2) = \frac{2 + 3}{2} = \frac{5}{2} = 2.5
\]
5. For \( x = 3 \):
\[
f^{-1}(3) = \frac{3 + 3}{2} = \frac{6}{2} = 3
\]

### Conclusion:
Comparing the computed values of \( f^{-1}(x) \) with the values given in the table:
- The values in the table do NOT match the output of our inverse function \( f^{-1}(x) \) derived from \( f(x) = 2x - 3 \).

So, the final responses are:

**Part (a)**:
\[
f^{-1}(x) = \frac{x + 3}{2}
\]

**Part (b)**:
The values for \( f^{-1} \) in the provided table do NOT fit the equation for \( f^{-1} \).