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Multiplying and dividing functions

See how we can multiply or divide two functions to create a new function.
Just like we can multiply and divide numbers, we can multiply and divide functions. For example, if we had functions f and g, we could create two new functions: fg and fg .

Multiplying two functions

Example

Let's look an example to see how this works.
Given that f(x)=2x3 and g(x)=x+1, find (fg)(x).

Solution

The most difficult part of combining functions is understanding the notation. What does (fg)(x) mean?
Well, (fg)(x) just means to find the product of f(x) and g(x). Mathematically, this means that (fg)(x)=f(x)g(x).
Now, this becomes a familiar problem.
(fg)(x)=f(x)g(x)Define.=(2x3)(x+1)Substitute.=2x2+2x3x3Distribute.=2x2x3Combine like terms.
Note: We simplified the result to obtain a nicer expression, but this is not necessary.

Let's try some practice problems.

Problem 1
c(y)=3y4d(y)=32y
Find (cd)(y).

Problem 2
m(x)=x23xn(x)=x5
Evaluate (mn)(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

Dividing two functions

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

Example

h(n)=2n1 and j(n)=n+3.
Let's find (jh)(n).

Solution

By definition, (jh)(n)=j(n)h(n).
We can now solve the problem.
(jh)(n)=j(n)h(n)Define.=n+32n1Substitute. 
Two important notes about this function:
  1. This function is simplified in its current form.
  2. The input n=12 is not a valid input for this function. This is because 2n1=0 at n=12, and division by 0 is undefined.

Let's try some practice problems

Problem 3
g(t)=t24h(t)=t+8
Find (gh)(t).

Problem 4
p(r)=5r2q(r)=r+2
Evaluate (pq)(4).
  • 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

Problem 5
f(x)=x+4g(x)=x3
For which value of x is (fg)(x) undefined?
  • 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

The distance and time that Jordan runs each day depends on the number of hours, h, that she works. The distance, D, in miles, and time, T, in minutes, that she runs are given by the functions D(h)=0.5h+8.5 and T(h)=6h+90, respectively.
Let function S represent the average speed at which Jordan runs on a day in which she works h hours.
Problem 6
Which of the following correctly defines function S?
Choose 1 answer:

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

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