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\(x\)
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Function Rules
and Output

Learn how to use a function rule to determine the output for any given input: plug in, compute, and verify.

ISEE Middle Level

What Is a Function?

A function is a rule that takes each input value and produces exactly one output value. Think of it as a machine: you feed a number in, the machine follows its rule, and a new number comes out.

Input
\(x\)
 Function Rule
\(f(x)\)
 Output
\(y\)

Input, Rule, and Output

Input (x)
The number you start with. It goes into the function.
Function Rule
The operation(s) you perform: add, subtract, multiply, divide, or combine.
Output (y)
The result after applying the rule. Written as \( f(x) \) or \( y \).

Why Do Functions Matter?

Functions appear everywhere: calculating sales tax, converting temperatures, predicting growth. On the ISEE, you will be given a rule and asked to find the output for a specific input, or identify the rule from a table of values.

Predict
Given any input, you can always find the output.
Organize
Input-output tables make patterns easy to spot.
Solve
Knowing the rule lets you work backward to find unknown inputs.

The Function Rule Formally

Common forms you will see on the ISEE:

1
Linear: \( f(x) = mx + b \) (multiply, then add)
2
Simple operations: \( f(x) = x^2 \), \( f(x) = \frac{x}{3} \), \( f(x) = x - 7 \)
3
Multi-step: \( f(x) = 3x^2 + 1 \) (power, multiply, then add)

Seeing a Rule in a Table

An input-output table shows several input-output pairs for the same rule. Here is the table for \( f(x) = 3x - 1 \):

Input \( x \) Rule: \( 3x - 1 \) Output \( f(x) \)
1\( 3(1) - 1 \)2
2\( 3(2) - 1 \)5
3\( 3(3) - 1 \)8
5\( 3(5) - 1 \)14

Expression vs. Function

These two ideas are closely related but not the same. The ISEE will test whether you know the difference.

Expression Function
What it isA math phrase: \( 2x + 5 \)A complete rule: \( f(x) = 2x + 5 \)
Has an = sign?NoYes
Gives output?Only when you evaluate itYes, for every input
Notation\( 2x + 5 \)\( f(x) = 2x + 5 \)

How to Find the Output

1
Read the rule. Identify the function: what operation(s) does it perform?
2
Substitute. Replace every \( x \) in the rule with the given input value.
3
Follow order of operations. Exponents first, then multiply/divide, then add/subtract.
4
State the output. Write your answer as \( f(\text{input}) = \text{result} \).

Example 1: \( f(x) = 4x - 7 \)

Find the output when \( x = 5 \).

Substitute
\(x=5\)
 Compute
\(4(5) - 7\)
 Simplify
\(20 - 7 = 13\)

Example 2: \( g(x) = x^2 + 2x \)

Find the output when \( x = -3 \).

1
Substitute: \( g(-3) = (-3)^2 + 2(-3) \)
2
Exponent first: \( (-3)^2 = 9 \)
3
Multiply: \( 2(-3) = -6 \)
4
Add: \( 9 + (-6) = \) 3

Example 3: From a Table

The table below follows the rule \( h(x) = \frac{x}{2} + 4 \). Verify and find \( h(10) \).

\( x \) \( h(x) \)
25
67
10?

Visualizing \( f(x) = 2x + 1 \)

x y 1 2 3 4 5 0 2 4 6 8 (0, 1) (1, 3) (2, 5) (3, 7) (4, 9)

Multi-Step Rules

Some function rules involve more than one operation. Always follow the order of operations (PEMDAS).

Square
\(4^2 = 16\)
 Multiply
\(2(16) = 32\)
 Subtract
\(3(4) = 12\)
 Combine
\(32-12+1=21\)

Working with Negative Inputs

Find \( f(-2) \) when \( f(x) = 5x + 3 \).

1
Substitute with parentheses: \( f(-2) = 5(-2) + 3 \)
2
Multiply: \( 5 \times (-2) = -10 \)
3
Add: \( -10 + 3 = \) \(-7\)

Common Mistakes to Avoid

✕ Forgetting Parentheses
Writing \( -3^2 = -9 \) instead of \( (-3)^2 = 9 \). Always wrap negative inputs in parentheses before applying exponents.
✕ Wrong Order of Operations
Adding before multiplying. In \( 2x + 3 \) with \( x = 5 \): do \( 2 \times 5 \) first, then add 3. Not \( 2 \times 8 \).
✕ Substituting the Output for the Input
If the question says "find \( f(3) \)," plug 3 in for \( x \), don't set the result equal to 3.

Summary: Function Rules at a Glance

Concept What to Remember
Function notation\( f(x) \) means "apply rule \( f \) to input \( x \)"
Finding outputReplace every \( x \) with the input, then simplify
Negative inputsAlways use parentheses: \( f(-n) = \text{rule}(-n) \)
Order of operationsExponents → Multiply/Divide → Add/Subtract
TablesEach row is one input-output pair; verify by plugging in

Question 1 of 5

If \( f(x) = 3x + 2 \), what is \( f(7) \)?

Question 2 of 5

If \( g(x) = x^2 - 4 \), what is \( g(-3) \)?

Question 3 of 5

A function follows the rule: multiply the input by 4, then subtract 5. What is the output when the input is 6?

Question 4 of 5

The table shows input-output pairs for a function. What is the missing output?

\( x \) \( f(x) \)
15
28
311
4?

Question 5 of 5

If \( f(x) = \frac{x}{3} + 7 \), what is \( f(12) \)?