CCSS Math: HSF.BF.A.1b
Become familiar with the idea that we can add, subtract, multiply, or divide two functions together to make a new function.
Just like we can add, subtract, multiply, and divide numbers, we can also add, subtract, multiply, and divide functions.

The sum of two functions

Part 1: Creating a new function by adding two functions

Let's add f(x)=x+1{f(x)=x+1} and g(x)=2x{g(x)=2x} together to make a new function.
Let's call this new function hh. So we have:
h(x)=f(x)+g(x)=3x+1{h(x)}={f(x)}+{g(x)}{=3x+1}

Part 2: Evaluating a combined function

We can also evaluate combined functions for particular inputs. Let's evaluate function hh above for x=2x=2. Below are two ways of doing this.
Method 1: Substitute x=2x=2 into the combined function hh.
h(x)=3x+1h(2)=3(2)+1=7\begin{aligned}h(x)&=3x+1\\\\ h(2)&=3(2)+1\\\\ &=\greenD{7} \end{aligned}
Method 2: Find f(2)f(2) and g(2)g(2) and add the results.
Since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2)+g(2)f(2) +g(2).
First, let's find f(2)f(2):
f(x)=x+1f(2)=2+1=3\begin{aligned}f(x)&= {x + 1}\\\\ f(2)&=2+1 \\\\ &=3\end{aligned}
Now, let's find g(2)g(2):
g(x)=2xg(2)=22=4\begin{aligned}g(x)&={2x}\\\\ g(2)&=2\cdot 2 \\\\ &=4\end{aligned}
So f(2)+g(2)=3+4=7f(2)+g(2)=3+4=\greenD7.
Notice that substituting x=2x =2 directly into function h h and finding f(2)+g(2)f(2) + g(2) gave us the same answer!

Now let's try some practice problems.

In problems 1 and 2, let f(x)=3x+2f(x)=3x+2 and g(x)=x3g(x)=x-3.

Problem 1

Find f(x)+g(x)f(x)+g(x).

Problem 2

Evaluate f(1)+g(1)f(-1)+g(-1).
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Here are a couple of ways to find the answer.
Method 1: Find f(1)f(-1) and g(1)g(-1) separately. Then add the two.
First, let's find f(1)f(-1):
f(x)=3x+2f(1)=3(1)+2=1\begin{aligned} f(x) &=3x+2\\\\ f(-1)&= 3(-1)+2\\\\ &= -1\end{aligned}
Next, let's find g(1)g(-1):
g(x)=x3g(1)=13=4\begin{aligned} g(x) &=x-3\\\\ g(-1)&= -1-3\\\\ &= -4\end{aligned}
So f(1)+g(1)=1+(4)=5f(-1)+g(-1)=-1+(-4)=\greenD{-5}
Method 2: Find f(x)+g(x)f(x)+g(x) first, and then evaluate this expression when x=1x=-1.
When x=1x=-1, this expression is equal to 4(1)14(-1)-1 or 5\greenD{-5}.

A graphical connection

We can also understand what it means to add two functions by looking at graphs of the functions.
The graphs of y=m(x)y=m(x) and y=n(x)y=n(x) are shown below. In the first graph, notice that m(4)=2m(4)=2. In the second graph, notice that n(4)=5n(4)=5.
Let p(x)=m(x)+n(x)p(x)=m(x)+n(x). Now look at the graph of y=p(x)y=p(x). Notice that p(4)=2+5=7p(4)=\blueD 2+\maroonD 5=\purpleD7.
Challenge yourself to see that p(x)=m(x)+n(x)p(x) = m(x) + n(x) for every value of xx by looking at the three graphs.

Let's practice.

Problem 3

The graphs of y=f(x)y=f(x) and y=g(x)y=g(x) are shown below.
Which is the best approximation of f(3)+g(3)f(3)+g(3)?
Choose 1 answer:
Choose 1 answer:

We can use the graphs of y=f(x)y=f(x) and y=g(x)y=g(x) to find f(3)f(3) and g(3)g(3).
The graph below shows that f(3)=6f(3)=6.
The graph below shows that g(3)=3g(3)=3.
So f(3)+g(3)=6+3=9f(3)+g(3)=6+3=9.

Other ways to combine functions

All of the examples we've looked at so far create a new function by adding two functions, but you can also subtract, multiply, and divide two functions to make new functions!
For example, if f(x)=x+3f(x)=x+3 and g(x)=x2g(x)=x-2, then we can not only find the sum, but also ...
... the difference.
f(x)g(x)=(x+3)(x2)       Substitute.=x+3x+2             Distribute negative sign.=5                                  Combine like terms.\begin{aligned}f(x)-g(x)&=(x+3)-(x-2)~~~~~~~\small{\gray{\text{Substitute.}}}\\\\ &=x+3-x+2~~~~~~~~~~~~~\small{\gray{\text{Distribute negative sign.}}}\\\\ &=5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Combine like terms.}}}\end{aligned}
... the product.
f(x)g(x)=(x+3)(x2)            Substitute.=x22x+3x6        Distribute.=x2+x6                   Combine like terms.\begin{aligned}f(x)\cdot g(x)&=(x+3)(x-2)~~~~~~~~~~~~\small{\gray{\text{Substitute.}}}\\\\ &=x^2-2x+3x-6~~~~~~~~\small{\gray{\text{Distribute.}}}\\\\ &=x^2+x-6~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Combine like terms.}}}\end{aligned}
... the quotient.
f(x)÷g(x)=f(x)g(x)=(x+3)(x2)                     Substitute.\begin{aligned}f(x)\div g(x)&=\dfrac{f(x)}{g(x)} \\\\ &=\dfrac{(x+3)}{(x-2)}~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Substitute.}}} \end{aligned}
In doing so, we have just created three new functions!

Challenge problem

p(t)=t+2p(t) = t + 2
q(t)=t1q(t) = t - 1
r(t)=tr(t) = t
Evaluate p(3)q(3)r(3)p(3)p(3) \cdot q(3) \cdot r(3) - p(3).
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let's first find p(3)p(3), q(3)q(3), and r(3)r(3) by substituting t=3t=3 into each of the formulas.
First, let's find p(3)p(3):
p(t)=t+2p(3)=3+2=5\begin{aligned}p(t) &=t+2\\\\ p(3)&= 3+2\\\\ &=\blueD5\end{aligned}
Next, let's find q(3)q(3):
q(t)=t1q(3)=31=2\begin{aligned} q(t) &=t-1\\\\ q(3)&= 3-1\\\\ &=\purpleC2\end{aligned}
Finally, let's find r(3)r(3):
r(t)=tr(3)=3=3\begin{aligned} r(t) &=t\\\\ r(3)&= 3\\\\ &=\goldD3\end{aligned}
We can now find p(3)q(3)r(3)p(3)p(3) \cdot q(3) \cdot r(3) - p(3) as follows:
p(3)q(3)r(3)p(3)=5235=305=25\begin{aligned} p(3) \cdot q(3) \cdot r(3) - p(3)&= \blueD5\cdot \purpleC2\cdot \goldD3-\blueD5 \\\\ &= 30-5 \\\\ &= 25 \end{aligned}
We could also find p(t)q(t)r(t)p(t)p(t) \cdot q(t) \cdot r(t) - p(t) first, and then find the value of this expression when t=3t=3. This would be more complicated, though, since the result would be a third degree polynomial!
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