In this chapter, the arithmetic of functions will be explored.  Two functions $f\left(x\right)$ and $g\left(x\right)$ having a common domain can be added, subtracted, multiplied, and divided, to form new functions$$

$h_1\left(x\right)\=f\left(x\right)\+g\left(x\right)$ 

$h_2\left(x\right)\=f\left(x\right)-g\left(x\right)$  

 $h_3\left(x\right)\=f\left(x\right)g\left(x\right)$     

$h_4\left(x\right)\=\frac{f\left(x\right)}{g\left(x\right)}$          

Our notation expresses the value of the functions at $x$, and thereby emphasizes the pointwise definitions used for the arithmetic of functions.

To prescribe one of the functions $h_k\left(x\right)$, we have to prescribe the recipe by which the values of $h_k\left(x\right)$ are to be computed. The notation shows, for example, that the value of $h_1\left(x\right)$ is simply the sum of the two numbers $f\left(x\right)$ and $g\left(x\right)$. This is the recipe at each point $x$ in the common domains of $f\left(x\right)$ and $g\left(x\right)$.

Similarly for the functions $h_2\left(x\right)$ and $h_3\left(x\right)$.  The only difficulty encountered in defining $h_4\left(x\right)$ is at zeros of $g\left(x\right)$ where the fraction $\frac{f\left(x\right)}{g\left(x\right)}$ would fail to be defined.  Thus, the domains for $h_k\left(x\right)\,k\=1\,2\,3$, will be the same as the common domain of $f\left(x\right)$ and $g\left(x\right)$, but the domain of $h_4\left(x\right)$ can be smaller.

The domain of $h_k\left(x\right)\,k\=1\,\mathrm{...}\,4$, cannot be larger than the common domain of $f\left(x\right)$ and $g\left(x\right)$.  For example, suppose the functions

$f\left(x\right)\=1\+\frac{1}{x}$ 

and

$g\left(x\right)\=2-\frac{1}{x}$ 

are added.  The rule for the sum would clearly be $h\left(x\right)\=3$, but the domain would not be all real numbers since the common domain for $f\left(x\right)$ and $g\left(x\right)$ was all the reals with the exception of $x\=0$.  That would still be the domain of the sum $h\left(x\right)$.

	Chapter Glossary
	$$ The following terms in Chapter 5 are linked to the Maple Math Dictionary. $$ axis of symmetry  domain  intercept  interval  midpoint  parabola quadratic formula  quotient  range  real number  zero  $$

Let $f\left(x\right)\=2x\+1$ and $g\left(x\right)\=x^2\+x-1$ be the rules for two functions whose common domain is the set of all real numbers.  In Problems 5.1 - 5.4, obtain the indicated arithmetic expression for the function $h\left(x\right)$, draw its graph, and determine its domain and range.  In addition, compute $h\left(3\right)$ and show that this value can also be obtained from the appropriate combination of the numbers $f\left(3\right)$ and $g\left(3\right)$.

5.1.  $h\left(x\right)\=f\left(x\right)\+g\left(x\right)$ 

5.2.  $h\left(x\right)\=f\left(x\right)-g\left(x\right)$

5.3.  $h\left(x\right)\=f\left(x\right)g\left(x\right)$

5.4.  $h\left(x\right)\=\frac{f\left(x\right)}{g\left(x\right)}$