# 10 Examples of Composite Functions with Answers

Composite functions are functions where we use the output values or results of one function as the inputs of another function. For example, if we have the functions f(x) and g(x), a composite function is formed when we write f(g(x)). Essentially, we are applying a function to the result of another function.

Here, we will look at 10 examples of compound functions. In addition, you will also be able to test your skills with 5 practice problems.

##### ALGEBRA

Relevant for

Learning about composite functions with examples.

See examples

##### ALGEBRA

Relevant for

Learning about composite functions with examples.

See examples

## 10 Examples of composite functions with answers

The following examples of composite functions have a detailed solution. However, try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

Find the value of $latex f(g(3))$ if we have the functions $latex f(x)=2x+5$ and $latex g(x)=x+6$.

We can solve this example by evaluating $latex g(3)$. Then, we have:

$latex g(3)=(3)+6$

$latex g(3)=9$

Now that we know the value of $latex g(3)$, we can use it in $latex f(g(3))$:

$latex f(g(3))=f(9)$

$latex =2(9)+5$

$latex f(g(3))=23$

### EXAMPLE 2

What is the value of $latex g(f(-3))$ if we have the functions $latex f(x)=x^2-5$ and $latex g(x)=2x-7$?

In this case, we have to evaluate in the opposite order compared to the previous examples. Therefore, we start by evaluating $latex f(-3)$:

$latex f(-3)=(-3)^2-5$

$latex f(-3)=4$

Using the value of $latex f(-3)$ in the composition $latex g(f(-3))$, we have:

$latex g(f(-3))=g(4)$

$latex =2(4)-4$

$latex g(f(-3))=4$

### EXAMPLE 3

If we have the functions $latex f(x)=2x+7$ and $latex g(x)=4x-6$, find the composite function $latex h(x)=f(g(x))$.

To find the composition of functions $latex f(g(x))$, we use the expression of function $latex g(x)$ as the input of the function $latex f(x)$. Therefore, we have:

$latex h(x)=f(g(x))$

$latex =f(4x-6)$

$latex =2(4x-6)+7$

$latex =8x-12+7$

$latex h(x)=8x-5$

### EXAMPLE 4

Find the composite function $latex h(x)=g(f(x))$ if $latex f(x)=2x^2+2$ and $latex g(x)=5x-4$.

In this case, to find the composition $latex g(f(x))$, we have to use the function $latex f(x)$ as the input of $latex g(x)$. Then, we have:

$latex h(x)=g(f(x))$

$latex =g(2x^2+2)$

$latex =5(2x^2+2)-4$

$latex =10x^2+10-4$

$latex h(x)=10x^2+6$

### EXAMPLE 5

Determine the value of $latex f(g(-4))$ if we have the functions $latex f(x)=x^2+2x+3$ and $latex g(x)=3x+8$.

To find the value of $latex f(g(-4))$, we have to start by evaluating $latex g(-4)$. Therefore, we have:

$latex g(-4)=3(-4)+8$

$latex g(-4)=-4$

Now, we have to use the value of g(-4) in the function f:

$latex f(g(-4))=f(-4)$

$latex =(-4)^2+2(4)+3$

$latex f(g(-4))=27$

### EXAMPLE 6

Find the composition $latex h(x)=f(g(x))$ with the functions $latex f(x)=-x^2+5x-10$ and $latex g(x)=x+2$.

The composition $latex f(g(x))$ is found by using the function $latex g(x)$ as the input to the function $latex f(x)$. Thus, we have:

$latex h(x)=f(g(x))$

$latex =f(x+2)$

$latex =-(x+2)^2+5(x+2)-10$

$latex =-x^2-4x-4+5x+10-10$

$latex h(x)=-x^2+x-4$

### EXAMPLE 7

If we have the function $latex f(x)=3x^2-20$, what is the value of $latex f(f(3))$?

In this case, we have the composition of a single function. This composition is similar to the previous ones, with the only difference that we use the same function twice. Therefore, we have:

$latex f(3)=3(3)^2-20$

$latex =3(3)^2-20$

$latex =3(9)-20$

$latex f(3)=7$

Now, we use the found value as the input of the function f:

$latex f(f(3)=f(7)$

$latex =3(7)^2-20$

$latex =3(49)-20$

$latex =147-20$

$latex =127$

### EXAMPLE 8

Find the composition $latex h(x)=g(g(x))$ if we have the function $latex g(x)=x^2-5$

We have a composition with the same function. However, to solve, we have to follow the same process.

The composition $latex g(g(x))$ is found by using the function $latex g(x)$ as the input of the same function $latex g(x)$. Therefore, we have:

$latex h(x)=g(g(x))$

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

$latex =(x^2-5)^2-5$

$latex =x^4-10x^2+25-5$

$latex h(x)=x^4-10x^2+20$

### EXAMPLE 9

If we have the functions $latex f(x)=3x+4$, $latex g(x)=5x-6$ and $latex h(x)=-x+4$, what is the value of $latex f(g(h(2)))$?

In this case, we have a composition of three functions. Therefore, we start by finding the value of $latex h(2)$:

$latex f(2)=-2+4$

$latex f(2)=2$

Now, we use the value of $latex f(2)$ to find the value of $latex g(h(2))$:

$latex g(h(2))=g(2)$

$latex =5(2)-6$

$latex =4$

Finally, we use the value of $latex g(h(2))$ in the function f:

$latex f(g(h(2)))=f(4)$

$latex =3(4)+4$

$latex =16$

### EXAMPLE 10

If we have the functions $latex f(x)=3x+4$, $latex g(x)=5x-6$ and $latex h(x)=-x+4$, find an expression for $latex i (x)=f(g(h(x)))$.

We start by finding an expression for $latex g(h(x))$:

$latex g(h(x))=5(-x+4)-6$

$latex =-5x+20-6$

$latex =-5x+14$

Now, we use $latex g(h(x))=-5x+14$ in the function f:

$latex i(x)=f(g(h(x)))$

$latex =f(-5x+14)$

$latex =3(-5x+14)+4$

$latex =-15x+42+4$

$latex i(x)=-15x+46$

## 5 Composite functions practice problems

Apply everything you have learned about composite functions to solve the following practice problems.