Question

For the function f defined by ​f(x)equals4xminus5​, complete the tables below for f and f Superscript negative 1.
Question content area bottom left
Part 1
Complete the table on the right.
.
.
.
Question content area bottom right
Part 1
x
​f(x)
x
f Superscript negative 1 Baseline left parenthesis x right parenthesis
negative 1
negative 9
negative 9
negative 1
0
  
enter your response here
negative 5
  
enter your response here
1
  
enter your response here
negative 1
  
enter your response here
2
  
enter your response here
3
  
enter your response here
3
  
enter your response here
7
  
enter your response here

Answer 1

Let's start with the function \( f(x) = 4x - 5 \).

### Part 1: Calculate \( f(x) \) values

We need to compute \( f(x) \) for different values of \( x \):

1. **For \( x = -1 \)**:
\[
f(-1) = 4(-1) - 5 = -4 - 5 = -9
\]

2. **For \( x = 0 \)**:
\[
f(0) = 4(0) - 5 = 0 - 5 = -5
\]

3. **For \( x = 1 \)**:
\[
f(1) = 4(1) - 5 = 4 - 5 = -1
\]

4. **For \( x = 2 \)**:
\[
f(2) = 4(2) - 5 = 8 - 5 = 3
\]

5. **For \( x = 3 \)**:
\[
f(3) = 4(3) - 5 = 12 - 5 = 7
\]

Now we can fill in the first table with these results:

| \( x \) | \( f(x) \) |
|------------|------------|
| -1 | -9 |
| 0 | -5 |
| 1 | -1 |
| 2 | 3 |
| 3 | 7 |

### Part 2: Find the inverse function \( f^{-1}(x) \)

To find \( f^{-1}(x) \), we need to solve the equation \( y = 4x - 5 \) for \( x \):

1. Replace \( y \) with \( x \):
\[
x = 4y - 5
\]

2. Rearranging gives:
\[
4y = x + 5
\]
\[
y = \frac{x + 5}{4}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x + 5}{4}
\]

### Part 3: Calculate \( f^{-1}(x) \) values

Now compute \( f^{-1}(x) \) for the corresponding values of \( x \) in the second table:

1. **For \( x = -9 \)**:
\[
f^{-1}(-9) = \frac{-9 + 5}{4} = \frac{-4}{4} = -1
\]

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

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

4. **For \( x = 3 \)**:
\[
f^{-1}(3) = \frac{3 + 5}{4} = \frac{8}{4} = 2
\]

5. **For \( x = 7 \)**:
\[
f^{-1}(7) = \frac{7 + 5}{4} = \frac{12}{4} = 3
\]

Now we can fill in the second table:

| \( x \) | \( f^{-1}(x) \) |
|------------|------------------|
| -9 | -1 |
| -5 | 0 |
| -1 | 1 |
| 3 | 2 |
| 7 | 3 |

With these calculations, we have filled in both tables successfully.