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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.

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 .