Comparing Functions — I can compare two functions using...

SECTION 1

Where Did the Idea of Comparing Functions Come From?

People have been comparing quantities for thousands of years. Ancient farmers needed to know which crop grew faster. Merchants wanted to figure out which trade route earned more money. Over time, mathematicians built tools — like functions — to describe these changing relationships with numbers.

A function is a rule that takes an input and gives exactly one output. When you have two functions, you can compare them to see which one produces bigger or smaller outputs. Let's look at a few key moments in the history of this idea.

~300 BCE

Euclid's Ratios

The Greek mathematician Euclid compared quantities using ratios. This was one of the earliest ways to say "this grows twice as fast as that."

1600s

Tables of Data

Scientists like Galileo recorded experiments in tables. By lining up two sets of data side by side, they could compare how different things changed over time.

1637

Descartes Invents the Coordinate Plane

René Descartes created the x-y coordinate plane. Now people could draw two functions on the same graph and instantly see which one was higher, lower, or growing faster.

1700s–Today

Modern Function Notation

Mathematicians like Euler introduced f(x) notation. Today we compare functions using equations like f(x) = 2x + 1 and g(x) = 3x − 4 to predict outputs without drawing anything.

The big question has always been the same: "If I change the input by the same amount for two functions, which output changes more?" That's exactly what you'll learn to answer in this lesson.

SECTION 2

Core Principles of Comparing Functions

Before you start comparing, you need a few foundational ideas. Think of these as your toolkit. Each tool helps you look at functions from a different angle.

Same Inputs, Different Outputs

To compare two functions fairly, plug in the same input values. Then look at the outputs. Whichever function gives a larger (or smaller) output at that input tells you something about how the functions relate.

Rate of Change

The rate of change tells you how much the output changes when you increase the input by 1. A function with a rate of change of 3 grows faster than one with a rate of change of 2.

Three Representations

Functions can be shown as graphs (pictures), tables (organized numbers), or equations (rules with variables). You can compare functions no matter which form they're in.

Starting Value (y-intercept)

The y-intercept is the output when the input is 0. Two functions can have the same rate of change but different starting values, or the same starting value but different rates of change.

✦ KEY TAKEAWAY

Think of two friends walking from different starting points at different speeds. The starting point is the y-intercept, and the speed is the rate of change. To compare their walks, you check where each person is at the same time — that's comparing outputs for equal inputs.

SECTION 3

Seeing Functions Side by Side on a Graph

A graph is one of the best ways to compare two functions. When you plot both lines on the same coordinate plane, you can instantly see which one is higher, which one is steeper, and where they might cross.

The cyan line shows f(x) = 2x + 1 and the violet line shows g(x) = x + 4. Notice how f(x) starts lower but climbs faster. The two lines cross at x = 3, where both outputs equal 7.

Look at the graph carefully. The cyan line (f) is steeper. That means its output grows by 2 for every 1 step in x. The violet line (g) is less steep — its output only grows by 1 for each step. Even though g starts higher (at 4 instead of 1), f eventually catches up and passes it.

💡 Quick Tip

On a graph, the steeper line has the greater rate of change. If two lines cross, the steeper one will always be higher after the crossing point.

SECTION 4

The Math Behind Comparing Functions

You don't always need a graph. You can compare functions using their equations. The key number to look at is the rate of change (also called the slope). For a linear function written as y = mx + b, the letter m is the rate of change and b is the starting value.

LINEAR FUNCTION FORM

y = mx + b

m = rate of change (how much y changes when x goes up by 1); b = y-intercept (the output when x = 0)

RATE OF CHANGE FROM A TABLE

rate of change = (y₂ − y₁) ÷ (x₂ − x₁)

Pick any two rows in a table. Subtract the y-values and divide by the difference in x-values. For example, if x goes from 1 to 2 and y goes from 5 to 8, the rate of change is (8 − 5) ÷ (2 − 1) = 3.

When you compare two functions, ask yourself two questions. First: which function has a greater rate of change? That function's output grows faster. Second: which function has a greater y-intercept? That function starts higher.

COMPARING TWO FUNCTIONS AT A SPECIFIC INPUT

Difference = f(x) − g(x)

Plug the same x-value into both functions. If f(x) − g(x) is positive, then f gives a larger output at that input. If it's negative, g gives a larger output.

SECTION 5

Comparing Functions Using Tables

Tables are super helpful because they line up inputs and outputs in neat rows. You can spot the rate of change by looking at how the output column changes from row to row. Let's compare two functions using a table.

Two side-by-side tables show the outputs for f(x) = 3x + 2 and g(x) = 5x − 1. The "Change" column shows how much the output increases each time x goes up by 1. Function g has the greater rate of change (+5 versus +3).

In the diagram above, both tables use the same x-values (0 through 4). The "Change" column is the key. For function f, the output always goes up by 3. For function g, the output always goes up by 5. That means g has a greater rate of change. Even though f starts higher (2 versus −1), g eventually catches up and passes f.

📋 How to Find Rate of Change in a Table

Look at any two rows where x increases by 1. Subtract the first y-value from the second y-value. That difference is the rate of change. If it's the same for every pair of rows, the function is linear (a straight line).

SECTION 6

Worked Example: Which Function Grows Faster?

Let's walk through a full example. You're given two functions in different forms, and you need to compare them.

❓ Problem

Function A is given as an equation: y = 4x + 3. Function B is given as a table: when x = 0, y = 7; when x = 1, y = 9; when x = 2, y = 11; when x = 3, y = 13. Which function has the greater rate of change? At what input do they have equal outputs?

Solution: Comparing Function A (equation) and Function B (table)

Step 1 — Find the rate of change for Function AFunction A is y = 4x + 3. This is in the form y = mx + b. The number in front of x is the rate of change.

Rate of change for A = 4

Step 2 — Find the rate of change for Function BLook at the table. When x goes from 0 to 1, y goes from 7 to 9. The change in y is 9 − 7 = 2. Check another pair: from x = 1 to x = 2, y goes from 9 to 11. The change is again 2.

Rate of change for B = 2

Step 3 — Compare the rates of changeFunction A's rate of change is 4. Function B's rate of change is 2. Since 4 > 2, Function A's output grows faster for each unit increase in x.

Function A has the greater rate of change.

Step 4 — Write Function B as an equationWe know the rate of change is 2 and when x = 0, y = 7. So the y-intercept is 7. Function B's equation is y = 2x + 7.

Function B: y = 2x + 7

Step 5 — Find where they are equalSet the equations equal: 4x + 3 = 2x + 7. Subtract 2x from both sides: 2x + 3 = 7. Subtract 3 from both sides: 2x = 4. Divide both sides by 2: x = 2. Plug x = 2 into either equation: y = 4(2) + 3 = 11.

They are equal at x = 2, where both outputs equal 11

SECTION 7

Graphs vs. Tables vs. Equations: Strengths and Limits

You now know three ways to compare functions. Each method has its own strengths. The table below summarizes when each one works best.

Comparing the three representations of functions

Method	Best For	Watch Out For
Graph	Seeing the big picture. You can quickly tell which function is higher, where they cross, and which one is steeper.	Hard to read exact values. If the lines are close together, it's tricky to tell them apart.
Table	Comparing exact output values. Great when a function is only given as data (no equation).	You only see the x-values listed. You can't see what happens between them or beyond them without more work.
Equation	Finding exact answers. You can plug in any x-value. You can also solve to find exactly where two functions are equal.	Requires some algebra skills. Not as immediately visual as a graph.

✦ KEY TAKEAWAY

Think of it like checking the weather. A graph is like looking out the window — you get the general picture fast. A table is like a weather report — it gives you specific numbers at specific times. An equation is like a weather formula — you can predict the temperature at any time, but you need to do some math.

SECTION 8

From Linear to Non-Linear: What Comes Next

So far, you've been comparing linear functions — functions whose graphs are straight lines and whose rate of change stays the same. In later math classes, you'll compare functions that curve, speed up, or slow down.

Linear vs. non-linear functions

Feature	Linear Functions (Now)	Non-Linear Functions (Later)
Graph Shape	Straight line	Curved (parabolas, exponentials, etc.)
Rate of Change	Constant (always the same)	Changes as x changes
Example	y = 3x + 1	y = x² or y = 2ˣ
Comparing	Compare m-values (slopes)	Compare outputs at several inputs; rate of change varies

The good news is that the skills you're learning now — reading graphs, building tables, and plugging into equations — work for all types of functions. You're building a foundation that will help you in Algebra, Geometry, and beyond.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Function A has a rate of change of 6 and a y-intercept of 2. Function B has a rate of change of 4 and a y-intercept of 10. Which function's output grows faster when the input increases? Which function starts with a higher output?

PROBLEM 2 — BASIC CALCULATION

Function P is y = 5x + 1. Function Q is y = 2x + 10. Find the output of each function when x = 3. Which function gives a larger output at x = 3?

PROBLEM 3 — INTERMEDIATE

Function M is shown in a table: x = 0, y = 5; x = 1, y = 8; x = 2, y = 11; x = 3, y = 14. Function N is given as y = 4x + 1. Find the rate of change of each function and determine at which x-value Function N's output first becomes greater than Function M's output.

PROBLEM 4 — APPLIED

Two friends start saving money. Amir has $12 in his jar and adds $5 each week. Bella has $30 in her jar and adds $3 each week. Write an equation for each person. After how many weeks will Amir have more money than Bella?

PROBLEM 5 — CRITICAL THINKING

Can two different linear functions have the same rate of change but never give the same output? If yes, give an example and explain what happens on a graph. If no, explain why not.

SUMMARY

Lesson Summary

To compare two functions, you can use graphs (look at steepness and where lines cross), tables (compare the output values row by row and check the constant change), or equations (compare the rate of change (m) and the y-intercept (b) in y = mx + b). A larger rate of change means the output grows faster for each unit increase in x.

When comparing, always use the same input values for both functions so the comparison is fair. You can find where two functions are equal by setting their equations equal and solving for x. The function with the greater rate of change will eventually have the larger output, even if the other function started higher. These skills prepare you for comparing more complex functions in algebra and beyond.