For the function f defined by ​f(x)equals4xminus5​, complete the tables below for f and f Superscript negative 1.

Question

Question content area bottom left
Part 1
Complete the table on the right.
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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.