Inverse Function Worksheets - Free Printable

Since I can't view or access images directly, I can't see the specific problem you're referring to. However, if your image contains a worksheet titled "Inverse of Functions" with tasks like finding the inverse of various functions (such as linear functions), I can help you solve those types of problems.

Let me assume the worksheet includes functions like:

1. $ f(x) = 4x - 3 $
2. $ f(x) = -3x + 17 $
3. $ f(x) = 8 + \frac{5}{x} $
4. $ f(x) = -2x - 6 $
5. $ f(x) = -4x - 7 $
6. $ f(x) = \frac{1}{2}x + \frac{1}{4} $
7. $ f(x) = -\frac{2}{3}x + 4 $
8. $ f(x) = \frac{10}{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) $.

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Let’s go through each one:

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

1. $ y = 4x - 3 $
2. Swap: $ x = 4y - 3 $
3. Solve:
$ x + 3 = 4y $ → $ y = \frac{x + 3}{4} $
4. $ f^{-1}(x) = \frac{x + 3}{4} $

✔ Answer: $ f^{-1}(x) = \frac{x + 3}{4} $

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

1. $ y = -3x + 17 $
2. Swap: $ x = -3y + 17 $
3. Solve:
$ x - 17 = -3y $ → $ y = \frac{17 - x}{3} $
4. $ f^{-1}(x) = \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 is not defined at $ x = 0 $. We must be careful.

1. $ y = 8 + \frac{5}{x} $
2. Swap: $ x = 8 + \frac{5}{y} $
3. Solve:
$ x - 8 = \frac{5}{y} $ → $ y = \frac{5}{x - 8} $
4. $ f^{-1}(x) = \frac{5}{x - 8} $

✔ Answer: $ f^{-1}(x) = \frac{5}{x - 8} $

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4. $ f(x) = -2x - 6 $

1. $ y = -2x - 6 $
2. Swap: $ x = -2y - 6 $
3. Solve:
$ x + 6 = -2y $ → $ y = -\frac{x + 6}{2} $
4. $ f^{-1}(x) = -\frac{x + 6}{2} $

✔ Answer: $ f^{-1}(x) = -\frac{x + 6}{2} $

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

1. $ y = -4x - 7 $
2. Swap: $ x = -4y - 7 $
3. Solve:
$ x + 7 = -4y $ → $ y = -\frac{x + 7}{4} $
4. $ f^{-1}(x) = -\frac{x + 7}{4} $

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

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6. $ f(x) = \frac{1}{2}x + \frac{1}{4} $

1. $ y = \frac{1}{2}x + \frac{1}{4} $
2. Swap: $ x = \frac{1}{2}y + \frac{1}{4} $
3. Solve:
$ x - \frac{1}{4} = \frac{1}{2}y $ → Multiply both sides by 2:
$ 2x - \frac{1}{2} = y $
4. $ f^{-1}(x) = 2x - \frac{1}{2} $

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

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7. $ f(x) = -\frac{2}{3}x + 4 $

1. $ y = -\frac{2}{3}x + 4 $
2. Swap: $ x = -\frac{2}{3}y + 4 $
3. Solve:
$ x - 4 = -\frac{2}{3}y $ → Multiply both sides by $-\frac{3}{2}$:
$ y = -\frac{3}{2}(x - 4) = -\frac{3}{2}x + 6 $
4. $ f^{-1}(x) = -\frac{3}{2}x + 6 $

✔ Answer: $ f^{-1}(x) = -\frac{3}{2}x + 6 $

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8. $ f(x) = \frac{10}{x} $

1. $ y = \frac{10}{x} $
2. Swap: $ x = \frac{10}{y} $
3. Solve: $ y = \frac{10}{x} $
4. $ f^{-1}(x) = \frac{10}{x} $

✔ Answer: $ f^{-1}(x) = \frac{10}{x} $

> Note: The function is its own inverse!

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✔ Final Answers Summary:

| Original Function | Inverse Function |
|-------------------|------------------|
| $ 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) = -2x - 6 $ | $ f^{-1}(x) = -\frac{x + 6}{2} $ |
| $ f(x) = -4x - 7 $ | $ f^{-1}(x) = -\frac{x + 7}{4} $ |
| $ f(x) = \frac{1}{2}x + \frac{1}{4} $ | $ f^{-1}(x) = 2x - \frac{1}{2} $ |
| $ f(x) = -\frac{2}{3}x + 4 $ | $ f^{-1}(x) = -\frac{3}{2}x + 6 $ |
| $ f(x) = \frac{10}{x} $ | $ f^{-1}(x) = \frac{10}{x} $ |

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If your image had different functions, please describe them or type them out, and I’ll solve them accordingly!