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.
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.
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.
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 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.
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).
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.
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.
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.