How to add functions and find the value of the sum at a specific point

What is the sum of two functions?

Did you know that you can add two different functions together?

There are two ways to add functions.

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

You can either input the value for $x$ into each function and then add the outputs together, or you can add the functions together and then input the value for $x$ and simplify. You can write the sum of functions as

$(f+g)(x)=f(x)+g(x)$

How to find the sum of two functions

---

Take the course

Want to learn more about Algebra 1? I have a step-by-step course for that. :)

---

Adding functions and then evaluating the resulting sum at a particular value

Example

Find $(f+g)(3)$ if $f(x)=x^2-x+4$ and $g(x)=x-2$.

We need to find $(f+g)(3)$, which we could rewrite as $(f)(3)+g(3)$. So we can plug $x=3$ into each function and then add the results together.

First let’s find $f(3)$.

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

Simplify using the order of operations.

$f(3)=9-3+4$

$f(3)=6+4$

$f(3)=10$

Now let’s find $g(3)$.

$g(3)=3-2$

$g(3)=1$

Now we can add the two together to find the sum.

$(f+g)(3)=f(3)+g(3)$

$(f+g)(3)=10+1$

$(f+g)(3)=11$

We also could have added the functions together and then plugged in $x=3$ to find the answer.

$(f+g)(x)=(x^2-x+4)+(x-2)$

Simplify by combining like terms.

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

Plug in $3$ for $x$.

$(f+g)(3)=3^2+2$

$(f+g)(3)=9+2$

$(f+g)(3)=11$

---

Let’s try another example of the sum of functions.

---

You can either input the value for x into each function and then add the outputs together, or you can add the functions together and then input the value for x and simplify.

Example

Find $(g+h)(-2)$ if $g(x)=x^2+5x$ and $h(x)=3-x$.

We need to find $(g+h)(-2)$, which we could rewrite as $(g)(-2)+h(-2)$. So we can plug $-2$ in for $x$ in each function and then add the results together.

First let’s find $g(-2)$.

$g(-2)=(-2)^2+5(-2)$

Simplify using the order of operations.

$g(-2)=4-10$

$g(-2)=-6$

Now let’s find $h(-2)$.

$h(-2)=3-(-2)$

$h(-2)=5$

Now we’ll find the sum of the functions at $x=-2$.

$(g+h)(-2)=g(-2)+h(-2)$

$(g+h)(-2)=-6+5$

$(g+h)(-2)=-1$

We also could have added the functions together and then plugged in $x=-2$ to find the answer.

$(g+h)(x)=(x^2+5x)+(3-x)$

Simplify by combining like terms.

$(g+h)(x)=x^2+4x+3$

Plug in $-2$ for $x$.

$(g+h)(-2)=(-2)^2+4(-2)+3$

$(g+h)(-2)=4-8+3$

$(g+h)(-2)=-4+3$

$(g+h)(-2)=-1$

---

Get access to the complete Algebra 1 course

---