Certain functions can be grouped together because they share a particular property. The property is often described symbolically using function notation.
For example, the functions given by $f(x)=x^2$f(x)=x2 and $g(x)=\cos x$g(x)=cosx share a symmetry property because $f(-x)=f(x)$f(−x)=f(x) and $g(-x)=g(x)$g(−x)=g(x). Any function whatsoever that has this property is called an even function.
Another symmetry property is expressed by the relation $f(-x)=-f(x)$f(−x)=−f(x). Functions with this property are called odd functions. They include the sine and tangent functions in trigonometry and all functions $h(x)=x^{2n-1}$h(x)=x2n−1 where $n$n is a positive integer.
Example 1
Verify that $h(x)=x^{2n-1}$h(x)=x2n−1 is an odd function when $n=1$n=1.
We have to show that $h(-x)=-h(x)$h(−x)=−h(x). Now, $h(x)=x^1$h(x)=x1. So, $h(-x)=(-x)^1=-(x^1)=-h(x)$h(−x)=(−x)1=−(x1)=−h(x), as required.
We could carry out similar steps in the case $n=2$n=2. This time $h(x)=x^3$h(x)=x3. Then, $h(-x)=(-x)^3=-(x^3)=-h(x)$h(−x)=(−x)3=−(x3)=−h(x) which shows that, again, $h$h is an odd function.
A property relating to addition
Some functions, $L$L, have the property $L(x+y)=L(x)+L(y)$L(x+y)=L(x)+L(y). For example, if $L(x)=3x$L(x)=3x, we have $L(x+y)=3(x+y)=3x+3y=L(x)+L(y)$L(x+y)=3(x+y)=3x+3y=L(x)+L(y). This property, together with the related property $L(ax)=aL(x)$L(ax)=aL(x) where $a$a is a constant, is called linearity.
Example 2
Show that the function $K(x)=3x+1$K(x)=3x+1 does not have the property $K(x+y)=K(x)+K(y)$K(x+y)=K(x)+K(y).
Since $K(x)=3x+1$K(x)=3x+1, we have $K(x+y)=3(x+y)+1$K(x+y)=3(x+y)+1. On expanding, we see that $K(x+y)=3x+3y+1$K(x+y)=3x+3y+1. But, $K(x)+K(y)=(3x+1)+(3y+1)$K(x)+K(y)=(3x+1)+(3y+1). So, the two are not equal.
Note that functions like $K(x)$K(x), where the highest power of $x$x is one, are called linear functions because their graphs are lines. They need not have the linearity property as described above, however.
A property involving a product
A property that has been very useful historically can be written $f(ab)=f(a)+f(b)$f(ab)=f(a)+f(b). The product $ab$ab might involve numbers with many decimal places and might, therefore, be tedious and time-consuming to calculate. The function $f$f turns the product into a sum. The sum can then be calculated easily and accurately and the original product is obtained by applying the inverse function $f^{-1}$f−1.
In symbols, this is
$ab=f^{-1}\left(f(ab)\right)=f^{-1}\left(f(a)+f(b)\right)$ab=f−1(f(ab))=f−1(f(a)+f(b)).
The logarithm function, discovered by Briggs and Napier, does just this and was essentially the way such calculations were done before hand-held computing devices were invented.
We know, for example, that $2^x\times2^y=2^{x+y}$2x×2y=2x+y. Suppose we wished to calculate the product $ab$ab. We could begin by expressing $a$a and $b$b separately as powers of $2$2. Thus, we put $a=2^x$a=2x and $b=2^y$b=2y, Then, $ab=2^x.2^y=2^{x+y}$ab=2x.2y=2x+y.
The process of obtaining the exponents $x$x from $a$a and $y$y from $b$b is the application of the base $2$2 logarithm function. For example, if $a=2^x$a=2x, we say $x$x is $\log_2(a)$log2(a).
To obtain the product $ab$ab from the sum $x+y$x+y, we apply the base $2$2 exponential function, which is the inverse of the logarithm function.
Example 3
It is easy to multiply $64$64 by $256$256 but we do it using the base $2$2 logarithm function by way of illustration.
The function maps $64$64 to $6$6, because $64=2^6$64=26.
The function maps $256$256 to $8$8, because $256=2^8$256=28.
Thus, $64\times256=2^6\times2^8=2^{14}=16384$64×256=26×28=214=16384.
Base $2$2 arithmetic is used in computing. Traditionally, logarithms with base $10$10 were used in routine calculations. In pure mathematics, logarithms with base $e\approx2.71828$e≈2.71828 have great importance. All the logarithm functions have the property $\log(ab)=\log(a)+\log(b)$log(ab)=log(a)+log(b).
Periodicity
We have seen in the trigonometric functions the property of periodicity. We know that, for example, $\tan x=\tan(x+n\pi)$tanx=tan(x+nπ), where $n$n is an integer, making the tangent function periodic with period $\pi$π.
In general, if $f(x+k)=f(x)$f(x+k)=f(x) for all values of $x$x, then $f$f is said to be periodic. If $k$k is the smallest constant for which this relation holds, we say the period is $k$k.
Another property for products
A function $g$g may have the property $g(xy)=g(x)g(y)$g(xy)=g(x)g(y). That is, for such a function, it makes no difference whether we multiply two numbers and then apply the function, or apply the function separately to the numbers and then multiply the results.
Example 4
Show that the square root function has the property $g(xy)=g(x)g(y)$g(xy)=g(x)g(y).
The function notation merely expresses the fact that $\sqrt{xy}=\sqrt{x}.\sqrt{y}$√xy=√x.√y.
This property is structurally similar to the linearity property mentioned before Example 2, but very different in effect.