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.

## 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 $latex g(x)=x+6$.

##### Solution

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

**EXAMPLE**

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

##### Solution

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:

$$g(x)=2x-7$$

$$g(4)=2(4)-7$$

$$g(4)=8-7$$

$$g(4)=1$$

**EXAMPLE **3

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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*:

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

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

$latex f(g(-4))=11$

**EXAMPLE **6

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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 $latex f(g(h(2)))$?

##### Solution

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

**EXAMPLE**

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

##### Solution

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.

## See also

Interested in learning more about algebraic functions? Take a look at these pages:

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