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Comparing Functions

There are many ways to represent functions. For example, a function can be represented with an input-output table , with a graph, and with an equation.

Sometimes a problem asks us to compare two functions which are represented in different ways. For example, you might be given a table and a graph, and asked which function is greater for a particular value, or which function increases faster.

Example :

Two functions are represented in different ways.

Function $1$ : The input-output table shows the $x$ - and $y$ -values of a quadratic function .

$x$	$y$
$0$	$0$
$1$	$1$
$2$	$4$
$3$	$9$
$4$	$16$
$6$	$36$
$8$	$64$

Function $2$ : The graph of a linear function is shown.

Math diagram

From the two functions, which function grows faster for large positive values of $x$ ?

In the graph, the $y$ -intercept is $5$ and the slope is $1$ . So, for $x = 0$ , the function shown in the graph has a greater value. Also, since the slope is positive, it's increasing.

However, if you look at the values in the table, you will see that the $y$ -values are equal to the square of $x$ . These values will have a faster-than-linear rate of growth.

For example, for the function in the table, when $x = 8$ , $y = 64$ . You can see in the graph that the line is not yet that high when $x = 8$ .