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Adding and subtracting 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 fg.

Adding two functions

Example

Let's look at an example to see how this works.
Given that f(x)=x+1 and g(x)=x22x+5, find (f+g)(x).

Solution

The most difficult part of combining functions is understanding the notation. What does (f+g)(x) mean?
Well, (f+g)(x) just means to find the sum of f(x) and g(x). Mathematically, this means that (f+g)(x)=f(x)+g(x).
Now, this becomes a familiar problem.
(f+g)(x)=f(x)+g(x)                             Define.=(x+1)+(x22x+5)        Substitute.=x+1+x22x+5                Remove parentheses.=x2x+6                                Combine like terms.

We can also see this graphically:

The images below show the graphs of y=f(x), y=g(x), and y=(f+g)(x).
From the first graph, we can see that f(2)=3 and that g(2)=5. From the second graph, we can see that (f+g)(2)=8.
So f(2)+g(2)=(f+g)(2) because 3+5=8.
Now you try it. Convince yourself that f(1)+g(1)=(f+g)(1).
Evaluate each expression.
f(1)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
g(1)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
(f+g)(1)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Let's try some practice problems.

In problems 1 and 2, let a(x)=3x25x+2 and b(x)=x2+8x10.

Problem 1

Find (a+b)(x).

Problem 2

Evaluate (a+b)(1).
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Subtracting two functions

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

Example

p(t)=2t1 and q(t)=t24t1.
Let's find (qp)(t).

Solution

Again, the most complicated part here is understanding the notation. But after working through the addition example, (qp)(t) means just what you'd think!
By definition, (qp)(t)=q(t)p(t). We can now solve the problem.
=(qp)(t)=q(t)p(t)Define.=(t24t1)(2t1)Substitute.=t24t12t+1Distribute negative sign.=t26tCombine like terms.
So (qp)(t)=t26t.

Let's try some practice problems.

Problem 3

j(n)=3n3n2+8
k(n)=8n2+3n5
Find (jk)(n).

Problem 4

g(x)=4x27x+2
h(x)=2x5
Evaluate (hg)(3).
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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(t)=526t and W(t)=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(t).
N(t)=

Challenge problem

The graphs of y=f(x) and y=g(x) are plotted on the grid below.
Which is the graph of y=(f+g)(x)?
Choose 1 answer:

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