Comparing
Functions
Learn to compare two functions using graphs, tables, and equations and describe how outputs change for equal input changes.
What Is a Function?
A function is a rule that assigns exactly one output to every input. We often write it as \( f(x) \), where \( x \) is the input and \( f(x) \) is the output.
Three Ways to Represent Functions
Functions can appear in different forms. To compare two functions, you need to read all three.
Why Compare Functions?
In real life you compare options all the time: two phone plans, two savings accounts, two speeds. Each option can be modeled by a function, and comparing their outputs tells you which is the better deal.
The Rate of Change
The rate of change tells you how much the output changes when the input changes by 1. For a linear function \( f(x) = mx + b \), the rate of change is the slope \( m \).
Comparing with Tables
When functions are given as tables, line up the same inputs and compare the outputs row by row.
| \( x \) | \( f(x) = 3x \) | \( g(x) = 2x + 2 \) |
|---|---|---|
| 0 | 0 | 2 |
| 1 | 3 | 4 |
| 2 | 6 | 6 |
| 3 | 9 | 8 |
| 4 | 12 | 10 |
Comparing with Graphs
On a graph, the steeper line has a greater rate of change. Where the lines cross, the functions are equal.
Comparing with Equations
When both functions are written as equations in slope-intercept form \( y = mx + b \), you can read two key features directly:
Initial Value vs. Rate of Change
These two features answer different questions. Mixing them up is the most common error when comparing functions.
| Feature | Initial Value \( b \) | Rate of Change \( m \) |
|---|---|---|
| Question it answers | Where does the function start? | How fast does the function grow? |
| On a graph | Where the line crosses the \( y \)-axis | Steepness of the line |
| In a table | Output when \( x = 0 \) | Change in output per row |
| Example | \( f(x)=2x+5 \) starts at 5 | \( f(x)=2x+5 \) grows by 2 |
How to Compare Two Functions
Example 1: Equation vs. Table
Function A: \( f(x) = 4x + 1 \). Function B: given by this table:
| \( x \) | \( g(x) \) |
|---|---|
| 0 | 3 |
| 1 | 6 |
| 2 | 9 |
| 3 | 12 |
Example 2: Two Graphs
Example 3: Mixed Representations
Function A is shown on a graph passing through \( (0, 5) \) and \( (4, 13) \). Function B: \( g(x) = 3x + 1 \). Which has a greater output at \( x = 6 \)?
Finding Where Functions Are Equal
Set the two function rules equal and solve for \( x \):
Common Mistakes
Summary Comparison
| What to compare | From an Equation | From a Table | From a Graph |
|---|---|---|---|
| Rate of Change | The coefficient \( m \) | \( \frac{\Delta y}{\Delta x} \) between rows | Steepness of the line |
| Initial Value | The constant \( b \) | Output when \( x = 0 \) | \( y \)-intercept |
| Where Equal | Set equations equal, solve | Same output in same row | Intersection point |
| Which Is Greater | Evaluate at the \( x \)-value | Compare outputs row by row | Which line is higher |
Question 1 of 5
\( f(x) = 5x + 2 \) and \( g(x) = 3x + 8 \). Which function has a greater rate of change?
Question 2 of 5
A table shows \( g(0)=4 \), \( g(1)=7 \), \( g(2)=10 \). What is the rate of change of \( g \)?
Question 3 of 5
\( f(x) = 2x + 6 \) and \( g(x) = 4x \). At which value of \( x \) are the two functions equal?
Question 4 of 5
On a graph, line \( f \) passes through \( (0, 1) \) and \( (3, 10) \). Line \( g \) passes through \( (0, 4) \) and \( (3, 7) \). Which function has a greater initial value?
Question 5 of 5
\( f(x) = x + 10 \) and \( g(x) = 3x + 2 \). Which statement is true?