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

Lesson 14: Combining functions

# 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: $f\cdot g$ and $\frac{f}{g}$ .

## Multiplying two functions

### Example

Let's look an example to see how this works.
Given that $f\left(x\right)=2x-3$ and $g\left(x\right)=x+1$, find $\left(f\cdot g\right)\left(x\right)$.

### Solution

The most difficult part of combining functions is understanding the notation. What does $\left(f\cdot g\right)\left(x\right)$ mean?
Well, $\left(f\cdot g\right)\left(x\right)$ just means to find the product of $f\left(x\right)$ and $g\left(x\right)$. Mathematically, this means that $\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$.
Now, this becomes a familiar problem.
$\begin{array}{rlr}\left(f\cdot g\right)\left(x\right)& =f\left(x\right)\cdot g\left(x\right)& \text{Define.}\\ \\ & =\left(2x-3\right)\cdot \left(x+1\right)& \text{Substitute.}\\ \\ & =2{x}^{2}+2x-3x-3& \text{Distribute.}\\ \\ & =2{x}^{2}-x-3& \text{Combine like terms.}\end{array}$
Note: We simplified the result to obtain a nicer expression, but this is not necessary.

## Let's try some practice problems.

Problem 1
$\begin{array}{rl}& c\left(y\right)=3y-4\\ \\ & d\left(y\right)=3-2y\end{array}$
Find $\left(c\cdot d\right)\left(y\right)$.

Problem 2
$\begin{array}{rl}& m\left(x\right)={x}^{2}-3x\\ \\ & n\left(x\right)=x-5\end{array}$
Evaluate $\left(m\cdot n\right)\left(-1\right)$.

## Dividing two functions

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

### Example

$h\left(n\right)=2n-1$ and $j\left(n\right)=n+3$.
Let's find $\left(\frac{j}{h}\right)\left(n\right)$.

### Solution

By definition, $\left(\frac{j}{h}\right)\left(n\right)=\frac{j\left(n\right)}{h\left(n\right)}$.
We can now solve the problem.
1. This function is simplified in its current form.
2. The input $n=\frac{1}{2}$ is not a valid input for this function. This is because $2n-1=0$ at $n=\frac{1}{2}$, and division by $0$ is undefined.

## Let's try some practice problems

Problem 3
$\begin{array}{rl}& g\left(t\right)={t}^{2}-4\\ \\ & h\left(t\right)=t+8\end{array}$
Find $\left(\frac{g}{h}\right)\left(t\right)$.

Problem 4
$\begin{array}{rl}& p\left(r\right)=5r-2\\ \\ & q\left(r\right)=r+2\end{array}$
Evaluate $\left(\frac{p}{q}\right)\left(4\right)$.

Problem 5
$\begin{array}{rl}& f\left(x\right)=x+4\\ \\ & g\left(x\right)=x-3\end{array}$
For which value of $x$ is $\left(\frac{f}{g}\right)\left(x\right)$ undefined?

## 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 $D\left(h\right)=-0.5h+8.5$ and $T\left(h\right)=-6h+90$, 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 $S$?

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 $y=\left(f\cdot g\right)\left(x\right)$?

## Want to join the conversation?

• I do not understand the graph equation
• You need to find the x values on both lines and multiply them together to find the value for the new graph of f*g(x). For example at x=4, g(4)=0 and f(4)=4 so f*g(4)=0 (multiply the two values together). When x=6, g(6)=-1 and f(6)=6 so f*g(6)=-6. With these two values, you know that the new graph will pass through the two points (4, 0) and (6,-6) which only happens in Graph B.
• in (t^2-4)/(t+8) why is simplified and we cannot resolve to t-0.5?
• this was very helpful 10/10
• The explanation for the last one doesn't make sense. Is there a simpler explanation?
• It does make sense....
You need to find points that correspond to (f*g)(x) = f(x) * g(x).
Sal creates a table of values for f(x) and g(x) from looking at the original graph.
Le'ts do one ...
Find f(4). If x = 4, then f(4) = 4
-- You find this by going right on the x-axis until you get to 4. Then, you go up until you hit the line that represents f(x). Then, you find the y-coordinate for this point.
Find g(4). If x = 4, then g(4) = 0
-- You find this similar to how you found f(4) except you find the point that is on the g(x) graph and find its y-coordinate.

Once you have f(4) and g(4), you can find f(4) * g(4) = 4 * 0 = 0
Thus the point for (f*g)(4) = (4, 0)

You can compare this point and other points to determine which graph to pick.
Hope this helps.
• Question one of both the multiplying and dividing questions seems to be missing the button that allows me to add an exponent to the problem
• Use the "carat" on the "6" key by pressing SHIFT at the same time you press the 6.
For example, x squared is x^2 and
x cubed is x^3, etc.
• why do we have to do this
• a(t)=(t−k)(t−3)(t−6)(t+3)a, left parenthesis, t, right parenthesis, equals, left parenthesis, t, minus, k, right parenthesis, left parenthesis, t, minus, 3, right parenthesis, left parenthesis, t, minus, 6, right parenthesis, left parenthesis, t, plus, 3, right parenthesis is a polynomial function of ttt, where kkk is a constant. Given that a(2)=0a(2)=0a, left parenthesis, 2, right parenthesis, equals, 0, what is the absolute value of the product of the zeros of aaa?
• 𝑡 = 2 ⇒ 𝑎(2) = (2 − 𝑘)(2 − 3)(2 − 6)(2 + 3) = 20(2 − 𝑘)

𝑎(2) = 0 ⇔ 20(2 − 𝑘) = 0 ⇔ 𝑘 = 2

So, 𝑎(𝑡) = (𝑡 − 2)(𝑡 − 3)(𝑡 − 6)(𝑡 + 3), which has zeros 𝑡 = −3, 𝑡 = 2, 𝑡 = 3 and 𝑡 = 6.

|(−3) ∙ 2 ∙ 3 ∙ 6| = 108
(1 vote)
• I'm just curious, but for the last question, what kind of function will lead me to those graph?
(1 vote)
• g(x) is the graph of sin, a trig function.