Question

Complete parts​ (a) and​ (b) below.
a. Write the inverse of ​f(x)equals3xminus5.
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
0
negative 5
negative 5
0
1
negative 2
negative 2
1
2
1
1
2
3
4
4
3
4
7
7
4
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.)

Answer 1

To find the inverse of the function \( f(x) = 3x - 5 \), we will follow these steps:

### Part (a)

1. **Replace \( f(x) \) with \( y \)**:
\[
y = 3x - 5
\]

2. **Swap \( x \) and \( y \)**:
\[
x = 3y - 5
\]

3. **Solve for \( y \)**:
- First, add 5 to both sides:
\[
x + 5 = 3y
\]
- Next, divide both sides by 3:
\[
y = \frac{x + 5}{3}
\]

4. **Write the inverse function**:
\[
f^{-1}(x) = \frac{x + 5}{3}
\]

So, the answer for part (a) is:
\[
f^{-1}(x) = \frac{x + 5}{3}
\]

### Part (b)

Now, we need to verify if the values in the table for \( f^{-1}(x) \) match the derived formula \( f^{-1}(x) = \frac{x + 5}{3} \).

Let's calculate \( f^{-1}(x) \) for the inputs given in the table and check if these match the provided \( f^{-1}(x) \) values.

| x (input) | f(x) | f inverse (given) | f inverse (calculated) |
|-----------|------------|---------------------|-------------------------|
| 0 | -5 | -5 | \( f^{-1}(0) = \frac{0 + 5}{3} = \frac{5}{3} \) |
| 1 | -2 | -2 | \( f^{-1}(1) = \frac{1 + 5}{3} = 2 \) |
| 2 | 1 | 1 | \( f^{-1}(2) = \frac{2 + 5}{3} = \frac{7}{3} \) |
| 3 | 4 | 4 | \( f^{-1}(3) = \frac{3 + 5}{3} = \frac{8}{3} \) |
| 4 | 7 | 7 | \( f^{-1}(4) = \frac{4 + 5}{3} = 3 \) |

Now we can summarize the comparison:

- For \( x = 0 \), \( f^{-1}(0) = \frac{5}{3} \) (not -5)
- For \( x = 1 \), \( f^{-1}(1) = 2 \) (not -2)
- For \( x = 2 \), \( f^{-1}(2) = \frac{7}{3} \) (not 1)
- For \( x = 3 \), \( f^{-1}(3) = \frac{8}{3} \) (not 4)
- For \( x = 4 \), \( f^{-1}(4) = 3 \) (not 7)

### Conclusion for Part (b)
The values for \( f^{-1}(x) \) provided in the table do not match the calculated values from the derived inverse function. Therefore, the answer for part (b) is:

The values for \( f^{-1}(x) \) in the provided table do not fit the equation \( f^{-1}(x) = \frac{x + 5}{3} \).