Inverse Function Worksheets - Free Printable

Since I can't view or access uploaded images directly, I can't see the specific problem you're referring to. However, based on your description, it appears that the task is to find the inverse of each given function.

Let me solve the problem assuming the functions listed in the image are:

1. $ f(x) = 4x - 3 $
2. $ f(x) = -3x + 17 $
3. $ f(x) = 8 + \frac{5}{x} $
4. $ f(x) = -20x + 6 $
5. $ f(x) = 4x - 7 $
6. $ f(x) = \frac{1}{3}x + \frac{1}{2} $
7. $ f(x) = -\frac{3}{4}x + 8 $
8. $ f(x) = \frac{19}{2}x $

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🔁 How to Find the Inverse of a Function:

To find the inverse of a function $ f(x) $, follow these steps:

1. Replace $ f(x) $ with $ y $.
2. Swap $ x $ and $ y $.
3. Solve for $ y $.
4. Replace $ y $ with $ f^{-1}(x) $.

Let’s go through each one.

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1. $ f(x) = 4x - 3 $

- Step 1: $ y = 4x - 3 $
- Step 2: $ x = 4y - 3 $
- Step 3: Solve for $ y $:
$$
x + 3 = 4y \Rightarrow y = \frac{x + 3}{4}
$$
- Answer: $ f^{-1}(x) = \frac{x + 3}{4} $

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2. $ f(x) = -3x + 17 $

- $ y = -3x + 17 $
- $ x = -3y + 17 $
- $ x - 17 = -3y \Rightarrow y = \frac{17 - x}{3} $
- Answer: $ f^{-1}(x) = \frac{17 - x}{3} $

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3. $ f(x) = 8 + \frac{5}{x} $

Note: This function has a restriction $ x \neq 0 $, so its inverse will too.

- $ y = 8 + \frac{5}{x} $
- $ x = 8 + \frac{5}{y} $
- $ x - 8 = \frac{5}{y} $
- $ y = \frac{5}{x - 8} $
- Answer: $ f^{-1}(x) = \frac{5}{x - 8} $, $ x \neq 8 $

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4. $ f(x) = -20x + 6 $

- $ y = -20x + 6 $
- $ x = -20y + 6 $
- $ x - 6 = -20y \Rightarrow y = \frac{6 - x}{20} $
- Answer: $ f^{-1}(x) = \frac{6 - x}{20} $

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5. $ f(x) = 4x - 7 $

- $ y = 4x - 7 $
- $ x = 4y - 7 $
- $ x + 7 = 4y \Rightarrow y = \frac{x + 7}{4} $
- Answer: $ f^{-1}(x) = \frac{x + 7}{4} $

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6. $ f(x) = \frac{1}{3}x + \frac{1}{2} $

- $ y = \frac{1}{3}x + \frac{1}{2} $
- $ x = \frac{1}{3}y + \frac{1}{2} $
- $ x - \frac{1}{2} = \frac{1}{3}y $
- Multiply both sides by 3: $ 3x - \frac{3}{2} = y $
- Answer: $ f^{-1}(x) = 3x - \frac{3}{2} $

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7. $ f(x) = -\frac{3}{4}x + 8 $

- $ y = -\frac{3}{4}x + 8 $
- $ x = -\frac{3}{4}y + 8 $
- $ x - 8 = -\frac{3}{4}y $
- Multiply both sides by $-\frac{4}{3}$: $ y = -\frac{4}{3}(x - 8) $
- $ y = -\frac{4}{3}x + \frac{32}{3} $
- Answer: $ f^{-1}(x) = -\frac{4}{3}x + \frac{32}{3} $

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8. $ f(x) = \frac{19}{2}x $

- $ y = \frac{19}{2}x $
- $ x = \frac{19}{2}y $
- $ y = \frac{2}{19}x $
- Answer: $ f^{-1}(x) = \frac{2}{19}x $

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✔ Final Answers:

| Function | Inverse |
|--------|--------|
| $ f(x) = 4x - 3 $ | $ f^{-1}(x) = \frac{x + 3}{4} $ |
| $ f(x) = -3x + 17 $ | $ f^{-1}(x) = \frac{17 - x}{3} $ |
| $ f(x) = 8 + \frac{5}{x} $ | $ f^{-1}(x) = \frac{5}{x - 8} $ |
| $ f(x) = -20x + 6 $ | $ f^{-1}(x) = \frac{6 - x}{20} $ |
| $ f(x) = 4x - 7 $ | $ f^{-1}(x) = \frac{x + 7}{4} $ |
| $ f(x) = \frac{1}{3}x + \frac{1}{2} $ | $ f^{-1}(x) = 3x - \frac{3}{2} $ |
| $ f(x) = -\frac{3}{4}x + 8 $ | $ f^{-1}(x) = -\frac{4}{3}x + \frac{32}{3} $ |
| $ f(x) = \frac{19}{2}x $ | $ f^{-1}(x) = \frac{2}{19}x $ |

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If the actual functions in your image differ from these, please describe them or list them here, and I’ll adjust the solution accordingly!