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 $x=8$.