Intro to combining functions

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

Problem 2

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.

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!

Esercizio impegnativo

p(t)=t+2p(t) = t + 2
q(t)=t1q(t) = t - 1
r(t)=tr(t) = t
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