Function Notation and Writing Linear Functions — SAT Math

Introduction

Function notation is the language the SAT uses to dress up linear equations. Questions in this skill — part of the Algebra domain — ask you to evaluate expressions like $3f(2) - f(-1)$ , solve $f(x) = k$ for $x$ , or write the rule $f(x) = mx + b$ from a couple of given facts. Mastering this unlocks a whole class of questions that look complicated but reduce to straightforward arithmetic once you see through the notation.

Core Concept

Reading f(x) notation

$f(x)$ is simply a name for the output when the input is $x$ . For a linear function, the rule always has the form:

f(x) = mx + b

where $m$ is the slope (rate of change) and $b$ is the $y$ -intercept (the output when $x = 0$ ).

Evaluating: Replace every $x$ with the given number.

f(x) = 4x - 3 \implies f(2) = 4(2) - 3 = 5

Multi-step expressions: Evaluate each piece separately, then combine.

3f(2) - f(-1) = 3(5) - [4(-1)-3] = 15 - (-7) = 22

The most common trap: $f(-1) = -7$ is negative, so subtracting it adds 7. Handle the signs carefully.

Solving for the input: Set the rule equal to $k$ and solve for $x$ — this is ordinary linear equation work (see the prerequisite note on Solving Linear Equations).

f(x) = 35 \implies 4x - 3 = 35 \implies x = \frac{38}{4} = 9.5

Writing the rule from given information

From two input/output pairs $(x_1, y_1)$ and $(x_2, y_2)$ :

m = \frac{y_2 - y_1}{x_2 - x_1}, \quad \text{then} \quad b = y_1 - m x_1

From one pair $(x_0, y_0)$ and a rate of change $m$ :

b = y_0 - m x_0

Both routes end at the same formula $f(x) = mx + b$ .

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Introduction

Core Concept

Reading f(x) notation

Writing the rule from given information

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