Function Notation and Composition

Part III: Functions

Functions are one of the ACT's reporting categories, and they show up in more places than the questions labeled "function." A function is a rule that takes an input and gives back exactly one output. Once you are comfortable reading that rule, the same habits carry you through graphs, models of real situations, sequences, and transformations. Part III builds those habits in four steps: function notation and composition (this unit), polynomial functions, other function families such as rational, exponential, and logarithmic functions, and finally transformations and piecewise functions.

Throughout this part, keep one idea in front of you: a function question is almost always asking you to substitute carefully. Most wrong answers come from substituting into the wrong place, working in the wrong order, or forgetting a restriction, not from hard math.

Function notation

Function notation $f(x)$ names a rule. Whatever replaces $x$ on the right side of the equation gets substituted everywhere $x$ appears. Evaluating a function at a specific input is direct substitution. If $f(x) = 3x - 4$, then $f(5) = 3(5) - 4 = 11$.

When the input is itself an expression, such as $f(a+2)$, every instance of $x$ in the rule gets replaced by that entire expression, not just the letter. With $f(x) = 3x - 4$:

$f(a+2) = 3(a+2) - 4 = 3a + 2$.

Put parentheses around the input as you substitute. This is the single best protection against sign errors. For $g(x) = x^2 + 2$, the value $g(-4)$ is $(-4)^2 + 2 = 16 + 2 = 18$. Without parentheses, $-4^2$ reads as $-16$, which is exactly the trap one of the wrong choices is built to catch.

Composition

Composition means evaluating one function, then feeding that result into another. The expression $f(g(x))$ always evaluates the inner function first: find $g(x)$, then substitute that number into $f$. The order matters. $f(g(x))$ and $g(f(x))$ are generally different, because composition is not commutative.

Composition, step by step

Worked example. Let $f(x) = x + 5$ and $g(x) = x^2$. Find $f(g(3))$.

Evaluate the inner function first.

$g(3) = 3^2 = 9$

$f(9) = 9 + 5 = 14$

Notice that reversing the order gives something different: $g(f(3)) = g(8) = 64$. A very common wrong answer on the ACT is the result of composing in the reverse order, so slow down and read which function is on the inside.

You will also see questions that hand you a composite as an equation to solve. For instance, if $f(x) = x^2$ and $g(x) = 2x - 1$, finding the values of $x$ for which $f(g(x)) = 9$ means solving $(2x-1)^2 = 9$. Take the square root of both sides, remembering both signs: $2x - 1 = 3$ or $2x - 1 = -3$, so $x = 2$ or $x = -1$. Forgetting the negative root is a classic way to lose the point.

Domain

The domain of a function is every input value that produces a valid output. On the ACT three conditions restrict a domain:

• A denominator cannot equal zero.
• The expression under an even root (a square root, for example) cannot be negative.
• The argument of a logarithm must be positive.

Each condition is solved as an inequality or an exclusion, not evaluated as if you were solving for a specific value. When a function combines more than one restriction, every condition must hold at once, so the domain is the intersection of all of them, not just the easiest one to spot.

Finding a function's domain

Worked example. Find the domain of $h(x) = \dfrac{\sqrt{x-3}}{x-8}$.

Two conditions apply at once: the radical and the denominator.

Radical: $x - 3 \ge 0$, so $x \ge 3$.

Denominator: $x - 8 \ne 0$, so $x \ne 8$.

Domain: $x \ge 3$ and $x \ne 8$.

Two details deserve attention. First, a square root allows zero, so the inequality is $\ge$, not $>$. A logarithm does not allow zero, so its inequality is strict. Second, when you solve a radical inequality such as $4 - x \ge 0$, moving the variable can flip the direction: $4 \ge x$ means $x \le 4$.

A composition can hide a domain restriction. For $f(x) = \sqrt{x}$ and $g(x) = x - 10$, the function $f(g(x)) = \sqrt{x - 10}$ needs the output of $g$ to satisfy $f$'s own requirement, so $x - 10 \ge 0$ and $x \ge 10$.

Range and context

The range is the set of output values a function can actually produce, the counterpart to domain: domain answers which $x$-values are allowed, range answers which $y$-values result. From a graph, read the domain along the $x$-axis and the range along the $y$-axis. Use open and closed endpoints, holes, and arrows to decide which boundary values are excluded and whether the range continues without bound. From a table, the range is simply the set of output values shown. For a transformed basic function, track what each transformation does to the original range: a vertical shift moves the whole range up or down, and a reflection over the $x$-axis flips its sign.

A real-world context can restrict the domain further than the algebra alone would suggest. The height of a ball is modeled by $h(t) = -16t^2 + 64t$. Algebraically, $t$ can be any real number, but the model only describes the ball while it is in the air. It lands when $-16t(t - 4) = 0$, which is $t = 4$, so the domain in context is $0 \le t \le 4$.

Where this appears as a model

Many word problems are function evaluation in disguise. If the temperature drops $3^\circ$F for every 1,000 feet of altitude starting from $68^\circ$F, the function $F(h) = 68 - 3h$ (with $h$ in thousands of feet) gives $F(4) = 56^\circ$F at 4,000 feet. Identify the input, identify the rule, and substitute.

Strategy summary

• Put parentheses around every substituted input, especially negatives.
• For $f(g(x))$, work inside out. Ask yourself which function is written on the inside.
• For domain, list every restriction (denominator, even root, logarithm) and require all of them together.
• Squared equations usually have two solutions. Check that you found both.
• When a model has a real-world setting, decide whether the context limits the inputs.