SOLUTION: Inverse Functions Worksheet - Studypool - Free Printable

Let's go through the Inverse Functions worksheet step by step and solve each problem with clear explanations.

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Section 1: Find the inverse, $ r $, of each relation $ s $.

To find the inverse of a relation, you switch the $ x $- and $ y $-values in each ordered pair. That is:
> If $ (x, y) \in s $, then $ (y, x) \in r = s^{-1} $

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#### Problem 1:
$ s = \{(1,0), (2,4), (4,6), (7,6), (9,10)\} $

Switch $ x $ and $ y $:

- $ (1,0) \rightarrow (0,1) $
- $ (2,4) \rightarrow (4,2) $
- $ (4,6) \rightarrow (6,4) $
- $ (7,6) \rightarrow (6,7) $
- $ (9,10) \rightarrow (10,9) $

So,
$$
r = \{(0,1), (4,2), (6,4), (6,7), (10,9)\}
$$

✔ Note: The inverse may not be a function if there are repeated $ x $-values (like two pairs with $ x=6 $). But here we just need the inverse relation.

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#### Problem 2:
$ s = \{(-9,0), (-3,2), (-5,4), (2,7), (-4,3)\} $

Switch $ x $ and $ y $:

- $ (-9,0) \rightarrow (0,-9) $
- $ (-3,2) \rightarrow (2,-3) $
- $ (-5,4) \rightarrow (4,-5) $
- $ (2,7) \rightarrow (7,2) $
- $ (-4,3) \rightarrow (3,-4) $

So,
$$
r = \{(0,-9), (2,-3), (4,-5), (7,2), (3,-4)\}
$$

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#### Problem 3:
Given a table:
| $ x $ | $ y $ |
|--------|--------|
| 1 | 5 |
| 2 | -2 |
| 3 | 1 |
| 4 | 0 |
| 5 | 1 |

To find the inverse, switch $ x $ and $ y $ values:

| $ x $ | $ y $ |
|--------|--------|
| 5 | 1 |
| -2 | 2 |
| 1 | 3 |
| 0 | 4 |
| 1 | 5 |

So, the inverse relation is:
$$
r = \{(5,1), (-2,2), (1,3), (0,4), (1,5)\}
$$

⚠️ Note: This inverse is not a function, because $ x=1 $ appears twice (with $ y=3 $ and $ y=5 $).

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#### Problem 4:
Given table:
| $ x $ | $ y $ |
|--------|--------|
| 0 | 1 |
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |

Switch $ x $ and $ y $:

| $ x $ | $ y $ |
|--------|--------|
| 1 | 0 |
| 4 | 1 |
| 7 | 2 |
| 10 | 3 |
| 13 | 4 |

So,
$$
r = \{(1,0), (4,1), (7,2), (10,3), (13,4)\}
$$

This is a function (each $ x $ is unique).

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Section 2: For each function $ f(x) $, write the equation of the inverse $ f^{-1}(x) $. Graph both on the coordinate plane.

We'll use algebra to find inverses. 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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#### Problem 5: $ f(x) = x - 3 $

Step-by-step:
- $ y = x - 3 $
- Swap: $ x = y - 3 $
- Solve: $ y = x + 3 $

So,
$$
f^{-1}(x) = x + 3
$$

✔ Graph:
- $ f(x) = x - 3 $: line with slope 1, y-intercept -3
- $ f^{-1}(x) = x + 3 $: line with slope 1, y-intercept 3
They are reflections over the line $ y = x $

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#### Problem 6: $ f(x) = \frac{1}{2}x + 5 $

- $ y = \frac{1}{2}x + 5 $
- Swap: $ x = \frac{1}{2}y + 5 $
- Solve:
$$
x - 5 = \frac{1}{2}y \\
2(x - 5) = y \\
y = 2x - 10
$$

So,
$$
f^{-1}(x) = 2x - 10
$$

✔ Graph:
- Original: slope $ \frac{1}{2} $, y-intercept 5
- Inverse: slope 2, y-intercept -10
Again, symmetric over $ y = x $

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Section 3: Use composition to determine whether $ f $ and $ g $ are inverse functions

Recall:
Two functions $ f $ and $ g $ are inverses if and only if:
- $ f(g(x)) = x $ AND $ g(f(x)) = x $

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#### Problem 7:
$ f(x) = x - 1 $, $ g(x) = x + 1 $

Check compositions:

1. $ f(g(x)) = f(x+1) = (x+1) - 1 = x $ ✔
2. $ g(f(x)) = g(x-1) = (x-1) + 1 = x $ ✔

✔ So, they are inverses.

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#### Problem 8:
$ f(x) = 4 - x $, $ g(x) = 4 - x $

Wait — both functions are the same?

Let’s check:

1. $ f(g(x)) = f(4 - x) = 4 - (4 - x) = 4 - 4 + x = x $ ✔
2. $ g(f(x)) = g(4 - x) = 4 - (4 - x) = x $ ✔

Even though they are the same function, they are their own inverses!

✔ So, they are inverses.

But note: this function is its own inverse — it's symmetric about $ y = x $.

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Summary of Answers

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#### Inverses of Relations:
1. $ r = \{(0,1), (4,2), (6,4), (6,7), (10,9)\} $
2. $ r = \{(0,-9), (2,-3), (4,-5), (7,2), (3,-4)\} $
3. $ r = \{(5,1), (-2,2), (1,3), (0,4), (1,5)\} $
4. $ r = \{(1,0), (4,1), (7,2), (10,3), (13,4)\} $

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#### Inverse Functions:
5. $ f^{-1}(x) = x + 3 $
6. $ f^{-1}(x) = 2x - 10 $

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#### Composition Check:
7. $ f $ and $ g $ are inverses ✔
8. $ f $ and $ g $ are inverses ✔

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✔ Final Notes:

- To find an inverse of a relation: swap $ x $ and $ y $.
- To find an inverse of a function: algebraically solve $ y = f(x) $ for $ x $, then swap.
- To verify inverses: check that $ f(g(x)) = x $ and $ g(f(x)) = x $
- The graphs of a function and its inverse are reflections across the line $ y = x $

Let me know if you'd like the graphs sketched or further explanation!