Understanding the algebra of functions begins with understanding what a function is and the conventions of function notation.
Definition
A function is a mapping from one set to another. That is, each element in a specified set of elements is matched with an element belonging to another set. The first set is called the domain of the function. The domain elements are mapped onto the image set, called the range.
There are various kinds of functions, depending among other things on the nature of the sets involved. In our work in this chapter, we assume that both sets are subsets of the real numbers.
We assume further that the functions we are concerned with will, for the most part, be continuous functions. Continuity in functions essentially means that two numbers in the domain will be mapped to numbers that are close together in the range provided the domain numbers were sufficiently close together. The graphs of continuous functions have no jumps or gaps.
A function can be defined differently on different intervals in the domain so that it may be continuous within certain intervals but not continuous overall.
In rigorous mathematical work, one always specifies the domain and range when defining a particular function but in less formal settings a function may be defined by a formula and we assume that the intended domain is the largest set of numbers for which the formula makes sense.
For example, the expression $\sqrt{x}$√x can define a function. For each number $x$x in the domain, the function maps $x$x to the number $\sqrt{x}$√x in the range. However, we recognise that negative numbers do not make sense for this formula and so we take the natural domain to be the non-negative real numbers.
Notation
Functions are labeled with letters from various alphabets. Often, we use $f$f or $g$g or $h$h to identify a particular function. The function label can be followed by a letter or number in brackets. This is the domain element that the function operates on. We might write $y=f(x)$y=f(x) meaning the element $x$x from the domain is mapped by the function $f$f to the range element $y$y.
We say $f(x)$f(x) is the value of the function $f$f at the point $x$x.
For example, if $f$f is the squaring function, so that $f(x)=x^2$f(x)=x2, then the expression $f(3)$f(3) has the value $9$9.
Functions can be added, subtracted, multiplied or divided when they have the same domains. If $f$f and $g$g are functions, we compute $f(x)+g(x)$f(x)+g(x) by adding the function values at $x$x for each function. The short-hand notation for function addition is $(f+g)(x)$(f+g)(x). In a similar way, we form $(f-g)(x)$(f−g)(x) for subtraction, $(fg)(x)$(fg)(x) for multiplication and $(f/g)(x)$(f/g)(x) for function division.
Another useful notation is that for function composition. If the range of a function $g$g is in the domain of a function $f$f, we can form $f\left(g(x)\right)$f(g(x)) - a function of a function. This can be written $(f\circ g)(x)$(f∘g)(x). with the intention that $g$g is applied to the element $x$x and then $f$f is applied to $g(x)$g(x).
Rational functions
Functions of the form $\frac{P(x)}{Q(x)}$P(x)Q(x), where $P$P and $Q$Q are polynomial functions, are called rational functions. The values of $x$x that make $Q(x)=0$Q(x)=0 must be excluded from the domain of the rational function since at these points the formula $\frac{P(x)}{Q(x)}$P(x)Q(x) has no meaning.
For example, the function given by $h(x)=\frac{x^2-3x+2}{x-1}$h(x)=x2−3x+2x−1 would take the indeterminate form $\frac{0}{0}$00 at $x=1$x=1 and so the natural domain of this function is $(-\infty,1)\cup(1,\infty)$(−∞,1)∪(1,∞), the union of two open intervals from which the number $1$1 is excluded.
A similar situation arises with other kinds of functions given by the quotient of two functions where the denominator function can take the value zero. A well-known example is the tangent function defined by $\tan x=\frac{\sin x}{\cos x}$tanx=sinxcosx. There are infinitely many values of $x$x for which $\cos x=0$cosx=0 and these are all missing from the domain of $\tan x$tanx.
Worked Examples
Question 1
Given the following values:
$f\left(2\right)=4$f(2)=4, $f\left(7\right)=14$f(7)=14, $f\left(9\right)=18$f(9)=18, $f\left(8\right)=16$f(8)=16
$g\left(2\right)=8$g(2)=8, $g\left(7\right)=28$g(7)=28, $g\left(9\right)=36$g(9)=36, $g\left(8\right)=32$g(8)=32
Find $\left(f+g\right)$(f+g)$\left(2\right)$(2)
question 2
If $f(x)=3x-5$f(x)=3x−5 and $g(x)=5x+7$g(x)=5x+7, find each of the following:
$(f+g)(x)$(f+g)(x)
$(f+g)$(f+g)$\left(4\right)$(4)
$(f-g)(x)$(f−g)(x)
$(f-g)$(f−g)$\left(10\right)$(10)
question 3
Let $f\left(x\right)=\frac{9}{x-7}$f(x)=9x−7 and $g\left(x\right)=\sqrt{x-2}$g(x)=√x−2.
What is the domain of $f\left(x\right)$f(x)?
$($($-\infty$−∞$,$,$\infty$∞$)$)
A$($($-\infty$−∞$,$,$0$0$)$)$\cup$∪$($($0$0$,$,$\infty$∞$)$)
B$($($-\infty$−∞$,$,$9$9$)$)$\cup$∪$($($9$9$,$,$\infty$∞$)$)
C$($($-\infty$−∞$,$,$7$7$)$)$\cup$∪$($($7$7$,$,$\infty$∞$)$)
DWhat is the domain of $g\left(x\right)$g(x)?
$($($-\infty$−∞$,$,$\infty$∞$)$)
A$($($-\infty$−∞$,$,$2$2$)$)$\cup$∪$($($2$2$,$,$\infty$∞$)$)
B$[$[$2$2$,$,$\infty$∞$)$)
C$($($2$2$,$,$\infty$∞$)$)
DWhat is the domain of the function $(ff)(x)$(ff)(x)?
$($($-\infty$−∞$,$,$7$7$)$)$\cup$∪$($($7$7$,$,$\infty$∞$)$)
A$[$[$2$2$,$,$\infty$∞$)$)
B$($($-\infty$−∞$,$,$9$9$)$)$\cup$∪$($($9$9$,$,$\infty$∞$)$)
C$($($-\infty$−∞$,$,$\infty$∞$)$)
DFind the function $(ff)(x)$(ff)(x):
What is the domain of the function $(f/g)(x)$(f/g)(x)?
$($($-\infty$−∞$,$,$7$7$)$)$\cup$∪$($($7$7$,$,$\infty$∞$)$)
A$($($2$2$,$,$7$7$)$)$\cup$∪$($($7$7$,$,$\infty$∞$)$)
B$($($2$2$,$,$\infty$∞$)$)
C$[$[$2$2$,$,$7$7$)$)$\cup$∪$($($7$7$,$,$\infty$∞$)$)
DFind the function $(f/g)(x)$(f/g)(x):