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## Algebra (all content)

### Course: Algebra (all content)>Unit 7

Lesson 14: Combining functions

See how we can add or subtract two functions to create a new function.
Just like we can add and subtract numbers, we can add and subtract functions. For example, if we had functions $f$ and $g$, we could create two new functions: $f+g$ and $f-g$.

### Example

Let's look at an example to see how this works.
Given that $f\left(x\right)=x+1$ and $g\left(x\right)={x}^{2}-2x+5$, find $\left(f+g\right)\left(x\right)$.

### Solution

The most difficult part of combining functions is understanding the notation. What does $\left(f+g\right)\left(x\right)$ mean?
Well, $\left(f+g\right)\left(x\right)$ just means to find the sum of $f\left(x\right)$ and $g\left(x\right)$. Mathematically, this means that $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$.
Now, this becomes a familiar problem.

### We can also see this graphically:

The images below show the graphs of $y=f\left(x\right)$, $y=g\left(x\right)$, and $y=\left(f+g\right)\left(x\right)$.
From the first graph, we can see that $f\left(2\right)=3$ and that $g\left(2\right)=5$. From the second graph, we can see that $\left(f+g\right)\left(2\right)=8$.
So $f\left(2\right)+g\left(2\right)=\left(f+g\right)\left(2\right)$ because $3+5=8$.
Now you try it. Convince yourself that $f\left(1\right)+g\left(1\right)=\left(f+g\right)\left(1\right)$.
Evaluate each expression.
$f\left(1\right)=$
$g\left(1\right)=$
$\left(f+g\right)\left(1\right)=$

## Let's try some practice problems.

In problems 1 and 2, let $a\left(x\right)=3{x}^{2}-5x+2$ and $b\left(x\right)={x}^{2}+8x-10$.

### Problem 1

Find $\left(a+b\right)\left(x\right)$.

### Problem 2

Evaluate $\left(a+b\right)\left(-1\right)$.

## Subtracting two functions

Subtracting two functions works in a similar way. Here's an example:

### Example

$p\left(t\right)=2t-1$ and $q\left(t\right)=-{t}^{2}-4t-1$.
Let's find $\left(q-p\right)\left(t\right)$.

### Solution

Again, the most complicated part here is understanding the notation. But after working through the addition example, $\left(q-p\right)\left(t\right)$ means just what you'd think!
By definition, $\left(q-p\right)\left(t\right)=q\left(t\right)-p\left(t\right)$. We can now solve the problem.
$\begin{array}{rl}& \phantom{=}\left(q-p\right)\left(t\right)\\ \\ & =q\left(t\right)-p\left(t\right)\phantom{\rule{1em}{0ex}}\text{Define.}\\ \\ & =\left(-{t}^{2}-4t-1\right)-\left(2t-1\right)\phantom{\rule{1em}{0ex}}\text{Substitute.}\\ \\ & =-{t}^{2}-4t-1-2t+1\phantom{\rule{1em}{0ex}}\text{Distribute negative sign.}\\ \\ & =-{t}^{2}-6t\phantom{\rule{1em}{0ex}}\text{Combine like terms.}\end{array}$
So $\left(q-p\right)\left(t\right)=-{t}^{2}-6t.$

## Let's try some practice problems.

### Problem 3

$j\left(n\right)=3{n}^{3}-{n}^{2}+8$
$k\left(n\right)=-8{n}^{2}+3n-5$
Find $\left(j-k\right)\left(n\right)$.

### Problem 4

$g\left(x\right)=4{x}^{2}-7x+2$
$h\left(x\right)=2x-5$
Evaluate $\left(h-g\right)\left(3\right)$.

## An application

One college states that the number of men, $M$, and the number of women, $W$, receiving bachelor degrees $t$ years since 1980 can be modeled by the functions $M\left(t\right)=526-t$ and $W\left(t\right)=474+2t$, respectively.
Let $N$ be the total number of students receiving bachelors degrees at that college $t$ years since 1980.
Write an expression for $N\left(t\right)$.
$N\left(t\right)=$

## Challenge problem

The graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ are plotted on the grid below.
Which is the graph of $y=\left(f+g\right)\left(x\right)$?

## Want to join the conversation?

• why is 4(-1)^2 = 4? Shouldn't it be 16 because -4x-4 = 16?
• Good question! You solve the exponent first because this comes before multiplication in the order of operations (BIDMAS) where...

B = Brackets (parentheses)
I = Index (exponent, power)
D = Division
M = Multiplication
S = Subtraction

Since I comes before M, you solve (-1)^2 = (-1) x (-1) = 1 first and then you multiply 4 x 1 so that the final answer is 4.
• Under the heading subtracting two functions isn't the final answer -t^2-6t?
• Yes, that is correct. The equation is correctly solved up until the final step when the negative coefficient is dropped when stating the final answer - that (q-p) (t ) = -t^2-6t.
• I do not understand any of the graph sections. They are all very confusing and I cannot seem to get it
• Please tell me exactly what you have trouble getting? The graph section shows what is going on visually when you combine the two functions. If you don't get why one function is graphed as a straight line and another as a curve (parabola) then I recommend finishing algebra basics & algebra 1 sections before starting algebra 2.
• The last question was pretty good. Can someone recommend any other questions like that?
• After adding a quadratic function and a linear function together like the first graph, why does it show new quadratic function, not linear function? I want to know how to show function after adding two different kinds of functions.
• The standard form of a quadratic equation is ax^2+bx+c. Note that this formula contains a linear equation in it - the bx. When you add a linear equation you're just adding to the bx of the quadratic function (and possibly, the c). The resulting equation will still be quadratic, since it will still contain an x^2. Therefore, its graph is still quadratic.
• In the first set of questions, they tell you to evaluate some expressions, but there is no graph until you click the help option. Can someone fix this please?
• Pranav, the two graphs above this question tell you what f(1), g(1), and (f+g)(1) are, just solve the questions with that.
Hope this helps
• In question 4 for Method 2 the equation is simplified first and THEN the values are added in which completely change the outcome. If you add in the value of x=3 first you get -60 not -16. That is very confusing because that is now how it was taught in previous examples. Why is it changed up with no explanation?
• ok so first you have to follow pemdas not bimdas and do the operations
(1 vote)
• 6-(5*3-(12)*16/(-8))
(1 vote)
• Follow order of operations rules (PEMDAS)
-- Do 5*3
-- Do -12 * 16, then divide that by -8
-- Subtract these 2 results
2) Last, do 6 - the result of the above steps.
Give it a try. comment back if you get stuck.