Read and use function notation, evaluate functions including composition, and identify domain and range from rules and graphs (Functions).

What this topic is asking

Function notation is the language the rest of the Functions area is written in. The ACT tests whether you can evaluate a function at a value, compose two functions, and read domain and range. These are mechanical once you read $f(x)$ correctly as "the output of $f$ when the input is $x$".

Reading and evaluating function notation

The single idea: $f(a)$ is a substitution, not a multiplication.

Putting the input in parentheses keeps the signs correct, especially for negative inputs.

Composition of functions

Composition feeds one function's output into another. $f(g(x))$ means: compute $g(x)$ first, then apply $f$ to that result. Order matters: $f(g(x))$ usually differs from $g(f(x))$. For $f(x) = x^{2}$ and $g(x) = x + 3$, $f(g(1)) = f(4) = 16$, while $g(f(1)) = g(1) = 4$. Always work from the innermost function outward.

Domain and range

The domain is every input the function allows; the range is every output it can produce.

• Exclude inputs that make a denominator zero (e.g. $\frac{1}{x - 2}$ excludes $x = 2$).
• Exclude inputs that make an even root negative (e.g. $\sqrt{x - 5}$ requires $x \ge 5$).
• For a polynomial, the domain is all real numbers.

The range is often read from the graph: the set of $y$-values the curve actually reaches. For $f(x) = x^{2}$, the range is $y \ge 0$ because a square is never negative.

Reading functions from graphs and tables

On a graph, $f(a)$ is the height of the curve above $x = a$: go up from $a$ on the $x$-axis to the curve, then across to the $y$-axis. To solve $f(x) = c$, find where the curve is at height $c$ and read the $x$-value(s). A table lists input-output pairs directly, so $f(a)$ is simply the output beside input $a$. Being able to move among rule, graph and table is exactly the fluency the ACT rewards, and it makes many questions a matter of reading rather than computing.

Solving for an input given an output

The reverse of evaluation is finding the input that yields a given output. If $f(x) = 2x - 7$ and you are told $f(x) = 5$, set the rule equal to the output and solve: $2x - 7 = 5$, so $2x = 12$ and $x = 6$. For a quadratic this can give two inputs, since a parabola reaches most heights at two points; for example $f(x) = x^{2}$ with $f(x) = 9$ gives $x = 3$ and $x = -3$. Reading the question carefully to see whether it wants an output (evaluate) or an input (solve) is essential, because the two run in opposite directions.

The vertical-line test

A relationship is a function only if each input has exactly one output. On a graph, this is the vertical-line test: if any vertical line crosses the curve more than once, the graph is not a function. A parabola opening up passes the test (one $y$ per $x$), but a sideways parabola or a full circle fails it, because some $x$-values map to two $y$-values. The ACT occasionally asks which graph "represents a function", and the vertical-line test answers it instantly.

Why notation fluency pays off

Function notation underlies linear, quadratic, exponential and every other function topic. Once $f(a)$ reliably means "substitute $a$", and composition means "inside out", the harder function questions reduce to the same substitution skill. The habit of writing the input in parentheses and reading a graph value carefully prevents the small errors that otherwise cost easy points.

Try this

Q1. If $f(x) = 5 - 2x$, find $f(-3)$. [1 point]

• Cue. $f(-3) = 5 - 2(-3) = 5 + 6 = 11$.

Q2. If $f(x) = x^{2}$ and $g(x) = x - 1$, find $g(f(3))$. [1 point]

• Cue. $f(3) = 9$, then $g(9) = 9 - 1 = 8$.