Free Printable Inverse Functions Worksheets for Students - Free Printable

Let's solve each problem step-by-step to find the inverse of each function. To find the inverse of a function $ f(x) $, we follow these general steps:

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

---

1) $ g(x) = -\frac{1}{x - 1} + 3 $

Step 1: Let $ y = -\frac{1}{x - 1} + 3 $

Step 2: Swap $ x $ and $ y $:
$ x = -\frac{1}{y - 1} + 3 $

Step 3: Solve for $ y $:
$$
x - 3 = -\frac{1}{y - 1}
$$
Multiply both sides by $ y - 1 $:
$$
(x - 3)(y - 1) = -1
$$
$$
y - 1 = \frac{-1}{x - 3}
$$
$$
y = 1 - \frac{1}{x - 3}
$$

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

---

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

Step 1: $ y = x - 6 $

Step 2: Swap: $ x = y - 6 $

Step 3: Solve: $ y = x + 6 $

✔ Answer: $ f^{-1}(x) = x + 6 $

---

3) $ g(x) = -\frac{2}{5}x - 2 $

Step 1: $ y = -\frac{2}{5}x - 2 $

Step 2: Swap: $ x = -\frac{2}{5}y - 2 $

Step 3: Solve:
$$
x + 2 = -\frac{2}{5}y
$$
$$
y = -\frac{5}{2}(x + 2)
$$

✔ Answer: $ g^{-1}(x) = -\frac{5}{2}(x + 2) $

---

4) $ f(x) = \sqrt[5]{x + 2} + 2 $

Note: $ \sqrt[5]{x+2} = (x+2)^{1/5} $

Step 1: $ y = (x + 2)^{1/5} + 2 $

Step 2: Swap: $ x = (y + 2)^{1/5} + 2 $

Step 3: Solve:
$$
x - 2 = (y + 2)^{1/5}
$$
Raise both sides to the 5th power:
$$
(x - 2)^5 = y + 2
$$
$$
y = (x - 2)^5 - 2
$$

✔ Answer: $ f^{-1}(x) = (x - 2)^5 - 2 $

---

5) $ g(x) = \frac{-4 + \sqrt[3]{4x}}{2} $

Note: $ \sqrt[3]{4x} = (4x)^{1/3} $

Step 1: $ y = \frac{-4 + (4x)^{1/3}}{2} $

Step 2: Swap: $ x = \frac{-4 + (4y)^{1/3}}{2} $

Step 3: Solve:
Multiply both sides by 2:
$$
2x = -4 + (4y)^{1/3}
$$
$$
2x + 4 = (4y)^{1/3}
$$
Cube both sides:
$$
(2x + 4)^3 = 4y
$$
$$
y = \frac{(2x + 4)^3}{4}
$$

✔ Answer: $ g^{-1}(x) = \frac{(2x + 4)^3}{4} $

---

6) $ f(x) = \frac{4}{5}x - 4 $

Step 1: $ y = \frac{4}{5}x - 4 $

Step 2: Swap: $ x = \frac{4}{5}y - 4 $

Step 3: Solve:
$$
x + 4 = \frac{4}{5}y
$$
$$
y = \frac{5}{4}(x + 4)
$$

✔ Answer: $ f^{-1}(x) = \frac{5}{4}(x + 4) $

---

7) $ h(n) = \frac{1}{n - 2} - 2 $

Step 1: $ y = \frac{1}{n - 2} - 2 $

Step 2: Swap: $ x = \frac{1}{y - 2} - 2 $

Step 3: Solve:
$$
x + 2 = \frac{1}{y - 2}
$$
$$
y - 2 = \frac{1}{x + 2}
$$
$$
y = 2 + \frac{1}{x + 2}
$$

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

---

8) $ g(x) = -\sqrt[5]{x} - 3 $

Note: $ \sqrt[5]{x} = x^{1/5} $

Step 1: $ y = -x^{1/5} - 3 $

Step 2: Swap: $ x = -y^{1/5} - 3 $

Step 3: Solve:
$$
x + 3 = -y^{1/5}
$$
$$
-(x + 3) = y^{1/5}
$$
Raise both sides to the 5th power:
$$
y = [-(x + 3)]^5 = -(x + 3)^5
$$

✔ Answer: $ g^{-1}(x) = -(x + 3)^5 $

---

9) $ g(x) = -2x^5 - 2 $

Step 1: $ y = -2x^5 - 2 $

Step 2: Swap: $ x = -2y^5 - 2 $

Step 3: Solve:
$$
x + 2 = -2y^5
$$
$$
y^5 = -\frac{x + 2}{2}
$$
$$
y = \left(-\frac{x + 2}{2}\right)^{1/5}
$$

✔ Answer: $ g^{-1}(x) = \left(-\frac{x + 2}{2}\right)^{1/5} $

---

10) $ f(x) = -\frac{1}{x} - 1 $

Step 1: $ y = -\frac{1}{x} - 1 $

Step 2: Swap: $ x = -\frac{1}{y} - 1 $

Step 3: Solve:
$$
x + 1 = -\frac{1}{y}
$$
$$
y = -\frac{1}{x + 1}
$$

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

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✔ Final Answers Summary:

| Problem | Inverse Function |
|--------|------------------|
| 1 | $ g^{-1}(x) = 1 - \frac{1}{x - 3} $ |
| 2 | $ f^{-1}(x) = x + 6 $ |
| 3 | $ g^{-1}(x) = -\frac{5}{2}(x + 2) $ |
| 4 | $ f^{-1}(x) = (x - 2)^5 - 2 $ |
| 5 | $ g^{-1}(x) = \frac{(2x + 4)^3}{4} $ |
| 6 | $ f^{-1}(x) = \frac{5}{4}(x + 4) $ |
| 7 | $ h^{-1}(x) = 2 + \frac{1}{x + 2} $ |
| 8 | $ g^{-1}(x) = -(x + 3)^5 $ |
| 9 | $ g^{-1}(x) = \left(-\frac{x + 2}{2}\right)^{1/5} $ |
| 10 | $ f^{-1}(x) = -\frac{1}{x + 1} $ |

Let me know if you'd like any explanation or verification for a specific one!