Use function notation and evaluate input–output pairs. | Linear functions | SAT Math

Function notation is just another way to write input-output relationships.

Core Idea

$𝑓 (𝑥)$ is just another name for $𝑦$ . To evaluate $𝑓 (3)$ , substitute 3 for $𝑥$ in the rule. To solve $𝑓 (𝑥) = 10$ , set the rule equal to 10 and solve for $𝑥$ .

Understanding

Function notation looks more formal than $𝑦 = 𝑚 𝑥 + 𝑏$ , but it works the same way. $𝑓 (𝑥) = 2 𝑥 + 5$ means: whatever you put in for $𝑥$ , double it and add 5.

Evaluating means going input → output. If $𝑓 (𝑥) = 2 𝑥 + 5$ , then $𝑓 (3) = 2 (3) + 5 = 11$ . On a graph, this is the $𝑦$ -coordinate at $𝑥 = 3$ . In a table, find the row where the input is 3 and read the output.

Solving goes the other direction: output → input. If $𝑓 (𝑥) = 20$ , then $2 𝑥 + 5 = 20$ , so $𝑥 = 7.5$ . On a graph, find where the horizontal line $𝑦 = 20$ meets the function.

The SAT may also use notation like $𝑔 (𝑡)$ or $𝑃 (𝑛)$ to reinforce that functions can use any variable names. Don't let unfamiliar letters throw you off — the process is identical.

Step by Step

To evaluate $𝑓 (𝑎)$ : replace every $𝑥$ in the function rule with $𝑎$ , then simplify.
To solve $𝑓 (𝑥) = 𝑘$ : set the function rule equal to $𝑘$ and solve for $𝑥$ .
To find $𝑓 (𝑎) - 𝑓 (𝑏)$ : evaluate each separately, then subtract.

Misconceptions

Treating $𝑓 (3)$ as $𝑓 \times 3$ — the parentheses indicate input, not multiplication.
Substituting into the wrong variable when the function uses a letter other than $𝑥$ , such as $𝑔 (𝑡) = 4 𝑡 - 1$ .
Confusing $𝑓 (𝑥) = 10$ (solve for $𝑥$ ) with $𝑓 (10)$ (plug in 10). The first asks "what input gives 10?" The second asks "what output does 10 give?"

Question

Worked Example

If $𝑓 (𝑥) = - 3 𝑥 + 14$ , what is the value of $𝑥$ when $𝑓 (𝑥) = 2$ ?

Select an answer to see the explanation