Linear functions

Official Digital SAT skill — Algebra domain.

What this question tests

Linear function questions on the Digital SAT test whether you can treat a function like a machine: you input a value, follow the rule exactly, and compute the output. They also test whether you can read the meaning of a linear model’s slope and y-intercept when the function describes a real situation, such as cost, distance, or temperature. You may see function notation like $f(x)$ and $g(x)$, expressions that combine outputs like $4f(2)-g(2)$, or a line described by a graph, a table, or a short context. The skill is tested because linear relationships are a foundation for modeling and because small errors in notation, substitution, or order of operations can completely change the result. If you can evaluate precisely and interpret consistently, you can handle many algebra and data questions with confidence.

What to know

A linear function has the form $f(x)=mx+b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept (the value when $x=0$).
Function notation $f(a)$ means “the output of the function $f$ when the input is $a$,” so you substitute $a$ everywhere $x$ appears in the rule.
When an expression uses a function output, such as $4f(2)$, you must evaluate $f(2)$ first and then multiply by $4$; the function value is a single number after substitution.
The slope $m$ represents the change in output per one-unit increase in input, often summarized as $m=\frac{\Delta y}{\Delta x}$ for a line.
The $y$-intercept $b$ is the starting value in many contexts, because it is the function value at $x=0$, so $f(0)=b$.
A linear model’s meaning depends on units: slope is “output units per input unit,” while the intercept is an output value at zero input.

How to approach it

Start by identifying exactly what is being asked, such as a single value like $f(3)$, a combined expression like $4f(2)-g(2)$, or an interpretation of slope or intercept, so you do not compute extra quantities you do not need.
Rewrite the function rule in your mind as a substitution instruction, then plug in the given input everywhere $x$ appears, which prevents leaving any $x$ unchanged.
Compute the inside function outputs before applying outside operations, because expressions like $4f(2)$ treat $f(2)$ as one number and order-of-operations mistakes are a common source of wrong answers.
If two functions appear, evaluate each function at the same input separately and label the results (for example, “$f(2)=...$” and “$g(2)=...$”), then combine them according to the given expression to avoid mixing rules.
For interpretation questions, identify the slope and intercept from the form $mx+b$ or from two points, and connect each to the context: slope is the rate, intercept is the value at zero input.
Check that your interpretation uses the correct units and wording, because “per” usually signals slope, while “starting value” or “base amount” usually signals intercept.

Common traps

Confusing slope and intercept happens because both are constants in $mx+b$, so students grab the first number they notice; avoid this by explicitly naming $m$ as the rate and $b$ as the value at $x=0$.
Misreading $4f(2)$ as $f(4\cdot 2)$ happens because the notation looks like multiplication; avoid it by remembering that the coefficient multiplies the output, not the input, so evaluate $f(2)$ first.
Treating $4f(2)$ as $4(2)+b$ happens when students substitute into the linear rule incorrectly; avoid it by substituting into the entire rule $f(x)=mx+b$ to get $f(2)=m\cdot 2+b$.
Combining values in the wrong order, like doing $g(2)-4f(2)$ instead of $4f(2)-g(2)$, happens when students rush; avoid it by rewriting the target expression clearly and following it left to right.
Choosing a “reasonable” number from context without tying it to slope or intercept happens because context feels intuitive; avoid it by stating what $x$ and $f(x)$ represent and then linking the slope to “per unit” and the intercept to $x=0$.

Tips & shortcuts

Box or underline the exact target expression and compute only what it requires to reduce distractions and arithmetic errors.
When interpreting a model, look for the word “per” to point to slope and the phrase “when $x=0$” to point to the intercept.
If a function is given as $mx+b$, quickly compute $f(0)=b$ and keep it in mind as the baseline value.
Label intermediate results, like $f(2)=9$ and $g(2)=14$, so combining them in expressions is straightforward and less error-prone.

Worked example

A linear function $h$ has slope $-3$ and satisfies $h(4)=5$. What is the value of $h(-2)$?

A. $23$
B. $-13$ ✓ (correct answer)
C. $33$
D. $-3$

Why: Use point-slope form with slope $-3$ through $(4,5)$: $h(x)-5=-3(x-4)$. Substitute $x=-2$: $h(-2)-5=-3(-6)=18$, so $h(-2)=23$. Therefore, the correct choice is C.

Use the Practice Questions for this skill to drill it, then attempt a Timed Practice Test.