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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: f, dot, g and start fraction, f, divided by, g, end fraction .

Multiplying two functions

Example

Let's look an example to see how this works.
Given that f, left parenthesis, x, right parenthesis, equals, 2, x, minus, 3 and g, left parenthesis, x, right parenthesis, equals, x, plus, 1, find left parenthesis, f, dot, 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, dot, g, right parenthesis, left parenthesis, x, right parenthesis mean?
Well, left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis just means to find the product of f, left parenthesis, x, right parenthesis and g, left parenthesis, x, right parenthesis. Mathematically, this means that left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, dot, g, left parenthesis, x, right parenthesis.
Now, this becomes a familiar problem.
(fg)(x)=f(x)g(x)Define.=(2x3)(x+1)Substitute.=2x2+2x3x3Distribute.=2x2x3Combine like terms.\begin{aligned} (f\cdot g)(x) &= f(x)\cdot g(x)&\gray{\text{Define.}} \\\\ &= \left(2x-3\right)\cdot\left(x+1\right) &\gray{\text{Substitute.}} \\\\ &= 2x^2+2x-3x-3&\gray{\text{Distribute.}} \\\\ &=2x^2-x-3&\gray{\text{Combine like terms.}} \end{aligned}
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\begin{aligned} &c(y)=3y-4 \\\\ &d(y)=3-2y \end{aligned}
Find left parenthesis, c, dot, d, right parenthesis, left parenthesis, y, right parenthesis.

Problem 2
m(x)=x23xn(x)=x5\begin{aligned} &m(x)=x^2-3x \\\\ &n(x)=x-5 \end{aligned}
Evaluate left parenthesis, m, dot, n, right parenthesis, left parenthesis, minus, 1, right parenthesis.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Dividing two functions

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

Example

h, left parenthesis, n, right parenthesis, equals, 2, n, minus, 1 and j, left parenthesis, n, right parenthesis, equals, n, plus, 3.
Let's find left parenthesis, start fraction, j, divided by, h, end fraction, right parenthesis, left parenthesis, n, right parenthesis.

Solution

By definition, left parenthesis, start fraction, j, divided by, h, end fraction, right parenthesis, left parenthesis, n, right parenthesis, equals, start fraction, j, left parenthesis, n, right parenthesis, divided by, h, left parenthesis, n, right parenthesis, end fraction.
We can now solve the problem.
(jh)(n)=j(n)h(n)Define.=n+32n1Substitute. \begin{aligned} \left(\dfrac{j}{h}\right)(n)&=\dfrac{j(n)}{h(n)}&\gray{\text{Define.}} \\\\ &= \dfrac{n+3}{2n-1}&\gray{\text{Substitute. }} \end{aligned}
Two important notes about this function:
  1. This function is simplified in its current form.
  2. The input n, equals, start fraction, 1, divided by, 2, end fraction is not a valid input for this function. This is because 2, n, minus, 1, equals, 0 at n, equals, start fraction, 1, divided by, 2, end fraction, and division by 0 is undefined.

Let's try some practice problems

Problem 3
g(t)=t24h(t)=t+8\begin{aligned} &g(t)=t^2-4 \\\\ &h(t)=t+8 \end{aligned}
Find left parenthesis, start fraction, g, divided by, h, end fraction, right parenthesis, left parenthesis, t, right parenthesis.

Problem 4
p(r)=5r2q(r)=r+2\begin{aligned} &p(r)=5r-2 \\\\ &q(r)=r+2 \end{aligned}
Evaluate left parenthesis, start fraction, p, divided by, q, end fraction, right parenthesis, left parenthesis, 4, right parenthesis.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 5
f(x)=x+4g(x)=x3\begin{aligned} &f(x)=x+4 \\\\ &g(x)=x-3 \end{aligned}
For which value of x is left parenthesis, start fraction, f, divided by, g, end fraction, right parenthesis, left parenthesis, x, right parenthesis undefined?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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, left parenthesis, h, right parenthesis, equals, minus, 0, point, 5, h, plus, 8, point, 5 and T, left parenthesis, h, right parenthesis, equals, minus, 6, h, plus, 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:
Choose 1 answer:

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, dot, g, right parenthesis, left parenthesis, x, right parenthesis?
Choose 1 answer:
Choose 1 answer:

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