Skip to content# Composite and Inverse Functions Examples With Worksheet

## What is an Inverse Function?

### Steps for Finding the Inverse of a Function

### Practice Questions

**Solution:**

## What is a Composite Function?

### Steps for Finding the Composite of a Function

### Practice Questions

**Solution:**

## Conclusion

## Worksheet on Composite and Inverse Functions

**Composite and Inverse Functions**

- Functions are mathematical rules that map one set of values (input) to another set of values (output). In mathematics, two types of functions are commonly used: inverse and composite functions.
- Understanding and applying these concepts enhance problem-solving skills and provide valuable tools for various fields of study.

In this article, we will discuss:

**What are Composite and Inverse Functions?****Steps for Finding the Composite and Inverse of a Function**

Here is one more link to practice a few extra questions: Maths Genie Composite and Inverse Functions Questions

- An inverse function is a function that reverses the output of another function.
- In other words, if we have a function f(x) that maps an input value x to an output value y, then the inverse function f⁻¹(x) maps the output value y back to the input value x.

- The inverse function of f is denoted by f⁻¹.

- To find the inverse of a function, follow these steps:

**Step #1: Replace f(x) with y.**

**Step #2: Interchange x and y.**

**Step #3: Solve for y in terms of x.**

**Step #4: Replace y with f⁻¹(x).**

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**Solved Example: **

**Question 1: Let f(x) = 3x + 4. To find the inverse of this function.**

Solution:

we follow the steps:

**Step #1:**Replace f(x) with y.

**y = 3x + 4**

**Step #2:**Interchange x and y.

**x = 3y + 4**

**Step #3:**Solve for y in terms of x.

**y = (x – 4)/3**

**Step #4:**Replace y with f⁻¹(x).

**f⁻¹(x) = (x – 4)/3**

Question 1: Determine the inverse of the function represented by f(x) = 4x - 3.

Answer :**Step #1:** Start with the given function

**f(x) = 4x - 3. **

**Step #2:** Represent the function as

**y = 4x - 3. **

**Step #3:** Exchange the positions of x and y in the equation, yielding

**x = 4y - 3. **

**Step #4:** Solve the equation for y:

**x + 3 = 4y, **

leading to the expression **y = (x + 3)/4. **

**Step #5:** Therefore, the inverse of f(x) is expressed as

**f⁻¹(x) = (x + 3)/4.**

Question 2: Consider the function g(x) = 2x - 5. Find the inverse of this function.

Answer :**Solution:**

**Step #1:** Start by replacing g(x) with y:

**y = 2x – 5**

**Step #2:** Swap the positions of x and y:

**x = 2y – 5**

**Step #3:** Solve for y in terms of x:

**y = x + 5/2**

**Step #4:** Replace y with g^{-1}(x):

**g ^{-1}(x) = x + 5/2**

- A composite function is a function that is obtained by combining two or more functions.
- The output of one function becomes the input of another function.
- The composite function is denoted by (f ∘ g)(x), which means “f of g of x”.

- To find the composite of a function, follow these steps:

**Step #1: Apply the first function to the input value.**

**Step #2: Take the output of the first function and apply the second function to it.**

**Step #3: Simplify the expression.**

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**Solved Example: **

**Question 1: ****Let f(x) = x² and g(x) = x – 1. ****To find (f ∘ g)(x)**

Solution:

we follow the steps:

**Step #1:**Apply g(x) to the input value.

**g(x) = x – 1.**

**Step #2:**Take the output of g(x) and apply f(x) to it.

**f(g(x)) = (x – 1)²**

**Step #3:**Simplify the expression.

**f(g(x)) = x² – 2x + 1.**

Question 1: Let f(x) = 2x - 1 and g(x) = x^{2} + 1. Find the composite function (g ∘ f)(x) and simplify it.

**Step #1:** Replace f(x) in g(f(x)):

**(g ∘ f)(x) = g(2x - 1).**

**Step #2:** Substitute f(x) into g(x):

**(g ∘ f)(x) = (2x - 1) ^{2} + 1 **

**= 4x ^{2} - 4x + 2.**

Question 2: Let f(x) = 2x + 3 and g(x) = x^{2} - 2x. Find the composite function (g ∘ f)(x) and simplify it.

**Solution:**

**Step #1:** Replace f(x) in g(f(x)):

**(g ∘ f)(x) = g(2x + 3).**

**Step #2:** Substitute f(x) into g(x):

**(g ∘ f)(x) = (2x + 3) ^{2} - 2(2x + 3)**

**= 4x ^{2} + 12x + 9 - 4x - 6**

**= 4x ^{2} + 8x + 3.**

- In conclusion, inverse and composite functions are fundamental concepts in mathematics that allow us to reverse the effect of a function and combine multiple functions to analyze complex relationships.
- Understanding and applying these concepts enhance problem-solving skills and provide valuable tools for various fields of study.
- Mastery of inverse and composite functions is essential for advancing in higher-level mathematics and related disciplines.

**Question 1:** Let f(x) = 3x + 2 and g(x) = 5x - 1. Find f(g(x)).

**Question 2:** Let f(x) = x² - 5x + 6 and g(x) = 2x + 1. Find f(g(x)).

**Question 3:** Let f(x) = 3x - 4. Find f⁻¹(x).

**Question 4:** Let f(x) = 2x - 1 and g(x) = 3x + 2. Find g(f(x)).