Applying Function Rules — Use a rule to find an output for...

SECTION 1

Where Did Functions Come From?

Have you ever used a vending machine? You press a button (your input), and a specific snack drops out (your output). Every time you press the same button, you get the same snack. That simple idea — one input gives exactly one output — is the heart of what mathematicians call a function. People have been thinking about this idea for hundreds of years.

~1670

Leibniz Coins the Word

German mathematician Gottfried Leibniz first used the word "function" to describe a math rule that connects one number to another.

1748

Euler Introduces f(x)

Leonhard Euler popularized writing functions as f(x). This notation is still the standard we use today in school and on the SHSAT.

1837

Dirichlet's Definition

Peter Dirichlet gave the modern definition: a function is any rule where each input has exactly one output. This is the definition you will use on the test.

Today

Functions on the SHSAT

Function questions appear regularly on the SHSAT. You are asked to apply a rule to find an output, complete a table, or identify a pattern.

The big question these mathematicians were trying to answer is: How can we write a clear rule that turns any input into the right output? That is exactly what you will learn to do in this lesson.

SECTION 2

Core Principles of Functions

Before you can apply a function rule, you need to understand four key ideas. These ideas come up again and again on the SHSAT.

Input (x)

The input is the number you start with. It is the value you plug into the rule. We often call it x.

Output (y or f(x))

The output is the number you get after the rule does its work. We write it as y or f(x).

Function Rule

The function rule tells you what to do to the input. For example, "multiply by 3, then add 1" is a rule. In symbols: f(x) = 3x + 1.

One Input → One Output

A function must give exactly one output for each input. If you put in 5, you always get the same answer — never two different answers.

✦ KEY TAKEAWAY

Think of a function like a recipe. The input is the ingredient you put in, the rule is the recipe steps, and the output is the finished dish. If you follow the same recipe with the same ingredient, you always get the same dish. That is how a function works — same input, same rule, same output.

SECTION 3

The Function Machine

One of the best ways to picture a function is as a function machine. You drop a number in the top, the machine applies the rule, and a new number comes out the bottom. The diagram below shows this for the rule f(x) = 2x + 3.

The left machine shows input x = 4 producing output 11. The right machine shows input x = −2 producing output −1. Both use the same rule: f(x) = 2x + 3.

Notice how the rule never changes. Only the input changes. When you plug in x = 4, you calculate 2 × 4 + 3 = 11. When you plug in x = −2, you calculate 2 × (−2) + 3 = −4 + 3 = −1. The machine does the same steps every time.

SECTION 4

The Math Behind Function Rules

On the SHSAT, you will see function rules written in a special way. Let's break down the notation (symbols) so nothing looks confusing on test day.

FUNCTION NOTATION

f(x) = expression involving x

f is the name of the function. x is the input variable. The expression on the right side tells you what to do with x.

EVALUATING A FUNCTION

f(3) means: replace every x with 3, then simplify

When you see f(3), do not multiply f times 3. Instead, substitute 3 for x everywhere in the rule.

EXAMPLE RULE

f(x) = 3x² − 5x + 2

To find f(4): replace x with 4 → 3(4)² − 5(4) + 2 → 3(16) − 20 + 2 → 48 − 20 + 2 = 30.

💡 SHSAT Tip

Always use parentheses when you substitute. Writing 3(4)² keeps things clear. Without parentheses, you might accidentally compute 34² = 1,156, which is very wrong!

Here is the step-by-step process you should always follow:

Read the rule. Identify the expression to the right of the equals sign.
Identify the input. Look for the number inside the parentheses (like the 3 in f(3)).
Substitute. Replace every x in the rule with the input. Use parentheses!
Simplify. Follow order of operations (PEMDAS) to get a single number.

SECTION 5

Input-Output Tables

The SHSAT often shows a table with several inputs and asks you to fill in the outputs. Sometimes it works the other way: you are given outputs and need to find the inputs. Let's see how to organize your work using an input-output table.

Input-output table for f(x) = 4x − 7

Input (x)	Rule: f(x) = 4x − 7	Output f(x)
0	4(0) − 7 = 0 − 7	−7
1	4(1) − 7 = 4 − 7	−3
2	4(2) − 7 = 8 − 7	1
3	4(3) − 7 = 12 − 7	5
5	4(5) − 7 = 20 − 7	13

Every row of the input-output table becomes a point on this graph. The input x goes along the horizontal axis, and the output f(x) goes up the vertical axis. Because the rule is linear (no exponents), the points form a straight line.

Notice how every time x goes up by 1, the output goes up by 4. That makes sense because the rule multiplies x by 4. The "−7" shifts everything down. Spotting this pattern can help you check your work quickly.

SECTION 6

Worked Example

Let's walk through an SHSAT-style problem from start to finish. Read every step carefully.

📝 PROBLEM

If f(x) = x² − 3x + 5, what is the value of f(−2)?

Step-by-Step Solution

Step 1 — Identify the Rule and InputThe rule is f(x) = x² − 3x + 5. The input is x = −2. We need to replace every x with −2.

Step 2 — Substitute with ParenthesesWrite: f(−2) = (−2)² − 3(−2) + 5. Notice: every x is replaced by (−2) inside parentheses. This is very important when the input is negative!

Step 3 — Evaluate the Exponent First (PEMDAS)(−2)² means (−2) × (−2) = 4. A negative times a negative is positive. So we now have: f(−2) = 4 − 3(−2) + 5.

(−2)² = 4

Step 4 — MultiplyNext, handle the multiplication: −3 × (−2) = 6. A negative times a negative is positive again. Now we have: f(−2) = 4 + 6 + 5.

−3(−2) = +6

Step 5 — Add Everything Up4 + 6 + 5 = 15. So f(−2) = 15.

f(−2) = 15

⚠️ WATCH OUT

The most common mistake is forgetting parentheses around negative numbers. Without them, −2² looks like −(2²) = −4 instead of (−2)² = +4. On the SHSAT, one sign error can lead to a wrong answer choice that the test makers put there on purpose!

SECTION 7

Common Mistakes vs. Correct Approaches

Many students lose points not because they don't understand functions, but because of small errors. Here are the most common mistakes and how to avoid them.

Common errors and how to fix them

Common Mistake	Why It's Wrong	Correct Approach
Treating f(3) as f × 3	f(3) is not multiplication. The parentheses mean "plug in 3."	Replace every x in the rule with 3, then simplify.
Writing −2² instead of (−2)²	−2² = −4 but (−2)² = 4. The parentheses change the answer.	Always wrap negative inputs in parentheses before applying exponents.
Forgetting order of operations	Adding before multiplying changes the result entirely.	Follow PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract.
Substituting only once when x appears twice	If the rule is x² + x, both x's must become the input.	Replace EVERY x in the expression with the input value.

✦ KEY TAKEAWAY

Think of substitution like a find-and-replace tool in a word processor. When you "find" the letter x and "replace" it with a number, you replace every single x, not just the first one. And you always put the replacement in parentheses to keep things safe.

SECTION 8

Connecting to Advanced Topics

Once you master applying a single function rule, the SHSAT and high school math will ask you to do more. Here is a preview of where this skill leads.

How today's skill connects to future math

What You Learn Now	What Comes Next
Plug one input into a rule → get one output	Plug the output of one function into another function (composition)
Fill in an input-output table	Discover the rule by looking at a table (reverse-engineering functions)
Work with rules like f(x) = 3x + 1 (linear)	Work with quadratic, exponential, and piecewise functions
Evaluate f(5) to get a number	Solve f(x) = 20 to find which input gives a target output

Don't worry about those advanced ideas right now. Just know that every single one of them starts with the skill you are building today: substitute, simplify, and find the output. Master this foundation, and the next steps will feel natural.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Write out every step — that's how you avoid careless errors on test day.

PROBLEM 1 — CONCEPTUAL

In your own words, explain what f(7) means if f(x) = 2x + 10. You do not need to calculate the answer — just describe the steps you would take.

PROBLEM 2 — BASIC CALCULATION

If g(x) = 5x − 8, find g(6).

PROBLEM 3 — INTERMEDIATE

If h(x) = x² + 4x − 1, find h(−3).

PROBLEM 4 — APPLIED

A taxi ride costs a flat fee of $3 plus $2 per mile. The total cost can be modeled by C(m) = 2m + 3, where m is the number of miles. How much does a 12-mile ride cost?

PROBLEM 5 — CRITICAL THINKING

If f(x) = 3x − 1 and you know that f(a) = 20, what is the value of a?

SUMMARY

Lesson Summary

A function is a rule that takes an input and produces exactly one output. When you see f(x), it means "apply the function rule to x." To evaluate a function, substitute the input for every x in the expression, always using parentheses around the value you plug in. Then simplify using PEMDAS (order of operations) to get a single number.

Remember the three-step process: read the rule, substitute the input, and simplify step by step. Watch out for negative inputs — always wrap them in parentheses to avoid sign errors. Build an input-output table when a problem gives you multiple inputs. This skill is the foundation for sequences, graphing, and every function topic you will see on the SHSAT and beyond.