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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, plus, g and f, minus, g.

### Example

Let's look at an example to see how this works.
Given that f, left parenthesis, x, right parenthesis, equals, x, plus, 1 and g, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x, plus, 5, find left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis.

### Solution

The most difficult part of combining functions is understanding the notation. What does left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis mean?
Well, left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis just means to find the sum of f, left parenthesis, x, right parenthesis and g, left parenthesis, x, right parenthesis. Mathematically, this means that left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, plus, g, left parenthesis, x, right parenthesis.
Now, this becomes a familiar problem.
\begin{aligned} (f+g)(x) &= f(x)+g(x) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Define.}}}\\\\ &= \left(x+1\right)+\left(x^2-2x+5\right) ~~~~~~~~\small{\gray{\text{Substitute.}}}\\\\ &= x+1+x^2-2x+5~~~~~~~~~~~~~~~~\small{\gray{\text{Remove parentheses.}}}\\\\ &=x^2-x+6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Combine like terms.}}} \end{aligned}

### We can also see this graphically:

The images below show the graphs of y, equals, f, left parenthesis, x, right parenthesis, y, equals, g, left parenthesis, x, right parenthesis, and y, equals, left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis.
From the first graph, we can see that f, left parenthesis, 2, right parenthesis, equals, start color #1fab54, 3, end color #1fab54 and that g, left parenthesis, 2, right parenthesis, equals, start color #11accd, 5, end color #11accd. From the second graph, we can see that left parenthesis, f, plus, g, right parenthesis, left parenthesis, 2, right parenthesis, equals, start color #e07d10, 8, end color #e07d10.
So f, left parenthesis, 2, right parenthesis, plus, g, left parenthesis, 2, right parenthesis, equals, left parenthesis, f, plus, g, right parenthesis, left parenthesis, 2, right parenthesis because start color #1fab54, 3, end color #1fab54, plus, start color #11accd, 5, end color #11accd, equals, start color #e07d10, 8, end color #e07d10.
Now you try it. Convince yourself that f, left parenthesis, 1, right parenthesis, plus, g, left parenthesis, 1, right parenthesis, equals, left parenthesis, f, plus, g, right parenthesis, left parenthesis, 1, right parenthesis.
Evaluate each expression.
f, left parenthesis, 1, right parenthesis, equals
g, left parenthesis, 1, right parenthesis, equals
left parenthesis, f, plus, g, right parenthesis, left parenthesis, 1, right parenthesis, equals

## Let's try some practice problems.

In problems 1 and 2, let a, left parenthesis, x, right parenthesis, equals, 3, x, squared, minus, 5, x, plus, 2 and b, left parenthesis, x, right parenthesis, equals, x, squared, plus, 8, x, minus, 10.

### Problem 1

Find left parenthesis, a, plus, b, right parenthesis, left parenthesis, x, right parenthesis.

### Problem 2

Evaluate left parenthesis, a, plus, b, right parenthesis, left parenthesis, minus, 1, right parenthesis.

## Subtracting two functions

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

### Example

p, left parenthesis, t, right parenthesis, equals, 2, t, minus, 1 and q, left parenthesis, t, right parenthesis, equals, minus, t, squared, minus, 4, t, minus, 1.
Let's find left parenthesis, q, minus, p, right parenthesis, left parenthesis, t, right parenthesis.

### Solution

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

## Let's try some practice problems.

### Problem 3

j, left parenthesis, n, right parenthesis, equals, 3, n, cubed, minus, n, squared, plus, 8
k, left parenthesis, n, right parenthesis, equals, minus, 8, n, squared, plus, 3, n, minus, 5
Find left parenthesis, j, minus, k, right parenthesis, left parenthesis, n, right parenthesis.

### Problem 4

g, left parenthesis, x, right parenthesis, equals, 4, x, squared, minus, 7, x, plus, 2
h, left parenthesis, x, right parenthesis, equals, 2, x, minus, 5
Evaluate left parenthesis, h, minus, g, right parenthesis, left parenthesis, 3, right parenthesis.

## 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 parenthesis, t, right parenthesis, equals, 526, minus, t and W, left parenthesis, t, right parenthesis, equals, 474, plus, 2, t, 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 parenthesis, t, right parenthesis.
N, left parenthesis, t, right parenthesis, equals

## Challenge problem

The graphs of y, equals, f, left parenthesis, x, right parenthesis and y, equals, g, left parenthesis, x, right parenthesis are plotted on the grid below.
Which is the graph of y, equals, left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis?