Function Notation and Evaluation

What $f(x)$ Actually Means

The notation $f(x)$ is one of the most misread symbols on the SAT. It does not mean $f$ multiplied by $x$. It means "the output of the function named $f$ when the input is $x$." Think of a function as a machine with a name: $f$ is the name of the rule, $x$ is what you drop into the machine, and $f(x)$ is what falls out the other end. If $f(x) = 3x + 2$, then $f$ is the rule "triple the input, then add two," and you can feed the machine anything — a number, another variable, or an entire expression.

This mental model matters because the SAT rarely asks you to state what a function is. It asks you to use one: evaluate it, read it off a graph, chain two of them together, or run it backward. Every one of those tasks is the same idea seen from a different angle.

Evaluating a Function

To evaluate a function, replace every $x$ in the rule with the given input, then simplify. Suppose $f(x) = x^2 - 4x + 1$. Then

$f(3) = (3)^2 - 4(3) + 1 = 9 - 12 + 1 = -2.$

Parentheses become critical the moment the input is negative:

$f(-2) = (-2)^2 - 4(-2) + 1 = 4 + 8 + 1 = 13.$

Two errors show up constantly here. The first is writing $(-2)^2 = -4$ instead of $4$ — squaring a negative gives a positive. The second is mishandling $-4(-2)$, which is $+8$, not $-8$. Wrapping the input in parentheses before you substitute prevents both.

Reading Functions from Graphs and Tables

A graph of $y = f(x)$ is a picture of every input–output pair at once. To find $f(5)$ from a graph, go to $x = 5$ on the horizontal axis, move straight up or down until you hit the curve, and read the height — that height is $f(5)$. To solve $f(x) = 0$, do the reverse: find where the curve crosses the $x$-axis, because the height there is zero.

Tables behave the same way. The row whose input column reads $5$ tells you $f(5)$ directly.

$x$	$f(x)$
$-1$	$4$
$0$	$1$
$2$	$-3$
$5$	$6$

From this table $f(2) = -3$, and the equation $f(x) = 4$ is solved by $x = -1$. The SAT loves to reverse the direction — handing you an output and asking for the input — so always check which column the question is pointing at before you answer.

Composition of Functions

Composition means using the output of one function as the input to another. The expression $f(g(x))$ says "first apply $g$, then apply $f$ to that result." Work strictly from the inside out.

Let $f(x) = 2x + 1$ and $g(x) = x^2$. To find $f(g(3))$:

$\begin{aligned} g(3) &= 3^2 = 9 \\ f(9) &= 2(9) + 1 = 19 \end{aligned}$

So $f(g(3)) = 19$. Reversing the order almost always changes the answer: $g(f(3)) = g(7) = 49$. Order is not interchangeable, so read the nesting carefully — the function on the inside runs first.

Running a Function Backward

Because a question can give you the output and ask for the input, you sometimes need to solve an equation rather than plug in a number. If $f(x) = 3x - 5$ and you are told $f(a) = 10$, set the rule equal to $10$ and solve: $3a - 5 = 10 \Rightarrow 3a = 15 \Rightarrow a = 5$. Recognizing when to substitute versus when to solve is half the battle on these items.

Domain and Range

The domain is the set of allowed inputs; the range is the set of possible outputs. On the SAT, domain restrictions come mostly from two sources: a denominator that cannot equal zero, and a square root whose contents cannot be negative. For $f(x) = \frac{x + 6}{x - 2}$, the input $x = 2$ is excluded because it makes the denominator zero, so the domain is every real number except $2$. Range is usually read from a graph — a parabola opening upward with its lowest point at $y = -3$ has range $y \geq -3$, since the curve never dips below that height.

Keep the two straight by anchoring them to the axes: domain lives on the horizontal ($x$) axis, range on the vertical ($y$) axis.