Free Printable Inverse Functions Worksheets for Students - Free Printable

Problem: Solve the tasks related to inverses of functions and domain/range.

#### Task 1: Given the ordered pairs \(\{(3, -2), (-1, 5), (4, 0)\}\)

Step 1: Find the inverse points.
To find the inverse of a relation, we swap the \(x\)-coordinates and \(y\)-coordinates of each ordered pair. The original ordered pairs are:
\[
\{(3, -2), (-1, 5), (4, 0)\}
\]
Swapping the coordinates, we get:
\[
\{(-2, 3), (5, -1), (0, 4)\}
\]

Step 2: Graph the inverse.
Plot the points \((-2, 3)\), \((5, -1)\), and \((0, 4)\) on the coordinate plane.

Step 3: Determine if this is an example of an inverse function.
A relation is a function if each \(x\)-value corresponds to exactly one \(y\)-value. In the inverse relation \(\{(-2, 3), (5, -1), (0, 4)\}\), each \(x\)-value \((-2, 5, 0)\) corresponds to exactly one \(y\)-value. Therefore, this is an example of an inverse function.

Answer for Task 1:
\[
\boxed{\{(-2, 3), (5, -1), (0, 4)\}}
\]

#### Task 2: Given the table

| \(x\) | 0 | 1 | 2 | 3 | 4 |
|-------|---|---|---|---|---|
| \(y\) | 3 | 1 | 2 | 4 | 2 |

Step 1: Find the inverse.
To find the inverse, swap the \(x\)-values and \(y\)-values from the table. The original table is:
\[
\begin{array}{c|c}
x & y \\
\hline
0 & 3 \\
1 & 1 \\
2 & 2 \\
3 & 4 \\
4 & 2 \\
\end{array}
\]
Swapping the values, we get:
\[
\begin{array}{c|c}
x & y \\
\hline
3 & 0 \\
1 & 1 \\
2 & 2 \\
4 & 3 \\
2 & 4 \\
\end{array}
\]
However, notice that the \(x\)-value 2 appears twice in the inverse table, which means it does not represent a function (since a function must have each \(x\)-value corresponding to exactly one \(y\)-value).

Step 2: Determine if this is an example of an inverse function.
Since the inverse relation has the \(x\)-value 2 appearing twice, it is not a function. Therefore, this is not an example of an inverse function.

Answer for Task 2:
\[
\boxed{
\begin{array}{c|c}
x & y \\
\hline
3 & 0 \\
1 & 1 \\
2 & 2 \\
4 & 3 \\
2 & 4 \\
\end{array}
}
\]
This is not an example of an inverse function.

#### Task 3: Given the graph

The graph shows a curve that passes the vertical line test, meaning it is a function. To find the inverse, we reflect the graph across the line \(y = x\).

Step 1: Reflect the graph across \(y = x\).
The reflected graph will be the inverse of the original function. The inverse graph will also pass the vertical line test, indicating it is a function.

Step 2: Determine if this is an example of an inverse function.
Since the reflected graph passes the vertical line test, it is a function. Therefore, this is an example of an inverse function.

Answer for Task 3:
The inverse is the reflection of the given graph across the line \(y = x\). This is an example of an inverse function.

#### Task 4: Verify that \(f\) and \(g\) are inverse functions. Find \(f(g(x))\) and \(g(f(x))\).

Given:
\[
f(x) = x + 2, \quad g(x) = x - 2
\]

Step 1: Compute \(f(g(x))\).
\[
f(g(x)) = f(x - 2) = (x - 2) + 2 = x
\]

Step 2: Compute \(g(f(x))\).
\[
g(f(x)) = g(x + 2) = (x + 2) - 2 = x
\]

Since \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f\) and \(g\) are inverse functions.

Answer for Task 4:
\[
\boxed{x}
\]

#### Task 5: Verify that \(f\) and \(g\) are inverse functions. Find \(f(g(x))\) and \(g(f(x))\).

Given:
\[
f(x) = 4x - 1, \quad g(x) = \frac{1}{4}x + \frac{1}{4}
\]

Step 1: Compute \(f(g(x))\).
\[
f(g(x)) = f\left(\frac{1}{4}x + \frac{1}{4}\right) = 4\left(\frac{1}{4}x + \frac{1}{4}\right) - 1 = x + 1 - 1 = x
\]

Step 2: Compute \(g(f(x))\).
\[
g(f(x)) = g(4x - 1) = \frac{1}{4}(4x - 1) + \frac{1}{4} = x - \frac{1}{4} + \frac{1}{4} = x
\]

Since \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f\) and \(g\) are inverse functions.

Answer for Task 5:
\[
\boxed{x}
\]

Final Answers:

1. \(\boxed{\{(-2, 3), (5, -1), (0, 4)\}}\)
2. \(\boxed{
\begin{array}{c|c}
x & y \\
\hline
3 & 0 \\
1 & 1 \\
2 & 2 \\
4 & 3 \\
2 & 4 \\
\end{array}
}\) (Not an inverse function)
3. The inverse is the reflection of the given graph across \(y = x\). (Yes, it is an inverse function)
4. \(\boxed{x}\)
5. \(\boxed{x}\)