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.
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
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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.
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???