Compare Properties of Functions Numerically

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  • 0:05 Functions Written Numerically
  • 1:10 Comparing Functions in Tables
  • 2:12 Comparing Different…
  • 3:23 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Can you compare functions when they're written out numerically as tables of values? In this lesson, you'll learn to do just that and to compare one function as a table to another written in a different form.

Functions Written Numerically

When most people think about functions, they think about function notation and equations that start with 'F(x)'. But F(x) is not actually what a function is; it's just one way to write a function. Conceptually, a function is a rule for transforming input values into output values. You have a certain range of input values, or x-values, that you can plug into the function. The function then tells you what you do to the x-values to transform them into output values, or y-values.

You can show functions in all kinds of ways, not just with function notation. Graphing is one alternative way, but you can also write a function numerically. A function written numerically is a table of values that shows the relationship between the inputs and the outputs of the function. Here's an example:

Example of function written numerically
table of values

You can probably guess here that the function is F(x) = 2x, since all the output values in the F(x) column are the input values multiplied by 2.

In this lesson, you'll learn to compare functions written in tables either to other functions written in tables or to functions represented in some other way.

Comparing Functions in Tables

If you are given two functions represented in tables, you can compare important properties of the functions, like their rates of change. You can do this either by looking at the tables directly or by converting both functions into a form you find easier, like a graph. Here's an example:

Function as a table of values
tables for functions

Given functions F and G, which function has the greater rate of change?

In this example, you could look at the two functions and see just from the numbers that G is increasing more quickly: as x goes from 1 to 5, F(x) goes from 2 to 10, while G(x) goes from 3 to 15. For the same change in x, F(x) increases by 8, while G(x) increases by 12. Therefore, G(x) has a greater rate of change.

If that doesn't make sense to you, you could also use the values given in the tables to draw some quick graphs of the functions. From these graphs below, it's clear that G(x) has a steeper upward slope, so G(x) must have a faster rate of change than F(x).

Function as a graph
graphs for example

Comparing Different Types of Functions

Sometimes, you'll also have to compare functions represented in two different forms. For example, you might have to compare the rate of change of a function written numerically in a table to a function drawn on a graph. Here's an example of that kind of comparison:

Comparing two functions
graph and table

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