Recall: Every object we've worked with has its own arithmetic:
- Numbers (Integers, Fractions)
- Polynomials
- Radical Expressions
- Rational Expressions
Definition: Let $f$ and $g$ be functions on the same domain $D$. Then:
We define the function $f+g$ by $(f+g)(x)=f(x)+g(x).$
We define the function $f-g$ by $(f-g)(x)=f(x)-g(x).$
We define the function $f \cdot g$ by $(f \cdot g)(x)=f(x) \cdot g(x).$
We define the function $\frac{f}{g}$ by $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}.$
Example: Consider the functions $f(x)=x^2+3$ and $g(x)=7x+5.$ Find $$ \begin{array}{l} (f+g)(x)\\ (f-g)(x)\\ (f \cdot g)(x)\\ (\frac{f}{g})(x)\\ \end{array} $$ Moreover, find $$ \begin{array}{l} (f+g)(-2)\\ (f-g)(-2)\\ (f \cdot g)(-2)\\ (\frac{f}{g})(-2)\\ \end{array} $$
Example
Consider the functions $f$ and $g$ defined by the following tables.
$\begin{array}{c|c}\hline x & f(x) \\ \hline -6 & 2\\ \hline -7 & 1\\ \hline 4 & -5\\ \hline\end{array}$ and $\begin{array}{c|c}\hline x & g(x) \\ \hline -6 & -3\\ \hline -7 & -8\\ \hline 4 & 0\\ \hline\end{array}$
Make the tables for the functions $f+g$ and $f \cdot g$.
Example
Consider the functions $f$ and $g$ defined as collections of ordered pairs.
$f=\{ (-8,3), (4,-1), (8,5)\}$ and $g=\{ (-8,6), (4,-3), (8,-9)\}$
Write the collection of ordered pairs which define $f-g$ and $\frac{f}{g}.$
Another Operation on Functions: Composition.
Suppose $g$ sends $-2$ to $3$, and $f$ sends $3$ to $5$. That is, in symbols $g(-2)=3$, and $f(3)=5$.
If we daisy-chain these two functions together, we get the composition: $$f(g(-2))=f(3)=5.$$ The above we may write as simply $$(f \circ g)(-2)=5.$$
Definition : Let $g$ be a function with domain $A$ and range $B$, and let $f$ be a function with domain $B$ and range $C$.
Then the composition of $f$ and $g$, denoted $f \circ g$, is defined to be $$(f \circ g)(x)=f(g(x))$$.
Example: : Consider the functions $f(x)=x^2−7$ and $g(x)=2x−4.$ Evaluate $(f \circ g)(2).$
Example: : Consider the functions $f(x)=x^2−7$ and $g(x)=2x−4.$ Evaluate $(f \circ g)(x).$
Warning: $f \circ g$ is generally NOT the same as $g \circ f.$
Example: : Consider the functions $f(x)=x^2−7$ and $g(x)=2x−4.$ Evaluate $(g \circ f)(2).$
Example: : Consider the functions $f(x)=x^2−7$ and $g(x)=2x−4.$ Evaluate $(g \circ f)(x).$
Example
Consider the functions $g$ and $f$ defined by the following tables.
$\begin{array}{c|c}\hline x & g(x) \\ \hline 9 & 2\\ \hline -5 & -6\\ \hline -7 & 0\\ \hline\end{array}$ and $\begin{array}{c|c}\hline x & f(x) \\ \hline 0 & 5\\ \hline 2 & 8\\ \hline -6 & 4\\ \hline\end{array}$
Write the table for the function $f \circ g.$
Fun Fact: Compositions of inverses always return $x$. In symbols $$(f \circ f^{-1})(x)=x \mbox{ and } (f^{-1} \circ f)(x)=x.$$ Example: Consider the tables $\begin{array}{c|c}\hline x & f^{-1}(x) \\ \hline 9 & 2\\ \hline -5 & -6\\ \hline -7 & 0\\ \hline\end{array}$ and $\begin{array}{c|c}\hline x & f(x) \\ \hline 2 & 9\\ \hline -6 & -5\\ \hline 0 & -7\\ \hline\end{array}$
Find the table for $f \circ f^{-1}.$
Example: Find the inverse of $f(x)=\sqrt[3]{x+4},$ then find $f \circ f^{-1}.$