Function notation questions look abstract, but they follow a small set of consistent rules. This guide explains how to read \(f(x)\) notation, how transformations shift and stretch graphs, and walks through two worked examples — one combining several transformations, one fully evaluating a composite function.
Reading Function Notation Correctly
\(f(x)\) simply means "the output of function \(f\) when the input is \(x\)." Evaluating \(f(3)\) means substituting \(3\) everywhere \(x\) appears in the function's rule. This sounds basic, but the SAT layers complexity on top of it — asking for \(f(x+2)\), \(f(g(x))\), or \(2f(x) - 1\) — and each of those means something specific and different.
- \(f(a)\): substitute \(a\) for every \(x\) in the rule and simplify.
- \(f(x) + k\): shifts the whole graph vertically by \(k\).
- \(f(x+h)\): shifts the whole graph horizontally, in the opposite direction of the sign of \(h\).
- \(f(g(x))\): a composite function — evaluate the inner function \(g(x)\) first, then feed that result into \(f\).
The Four Core Transformations
Every transformation question on the SAT is a variation of one of these four moves applied to a parent function like \(f(x)=x^2\):
| Transformation | Rule | Effect on the graph |
|---|---|---|
| Vertical shift | \(f(x) + k\) | Moves up (\(k>0\)) or down (\(k<0\)) |
| Horizontal shift | \(f(x-h)\) | Moves right (\(h>0\)) or left (\(h<0\)) |
| Vertical stretch/compress | \(a \cdot f(x)\) | Stretches (\(|a|>1\)) or compresses (\(0<|a|<1\)) |
| Reflection | \(-f(x)\) or \(f(-x)\) | Flips over the \(x\)-axis or \(y\)-axis |
The horizontal shift is the one students consistently get backwards — \(f(x-3)\) shifts right, not left, because it takes an input 3 larger to produce the same output the original function gave at \(x\).
Worked Example: Combined Transformation
Given \(f(x) = x^2\), describe the graph of \(g(x) = -2f(x-1) + 3\) and find \(g(4)\).
- Identify each transformation piece by piece: the \(-2\) outside stretches vertically and reflects over the \(x\)-axis; the \((x-1)\) shifts right by 1; the \(+3\) shifts up by 3.
- Substitute the definition of \(f\): \(g(x) = -2(x-1)^2 + 3\).
- To find \(g(4)\), substitute \(x=4\): \(g(4) = -2(4-1)^2 + 3\).
- Simplify inside the parentheses: \(g(4) = -2(3)^2 + 3\).
- Square first: \(g(4) = -2(9) + 3\).
- Multiply, then add: \(g(4) = -18 + 3 = -15\).
Notice the order of operations discipline: parentheses, then the exponent, then multiplication, then addition — the same order every time, regardless of how complicated the function looks.
Worked Example: Evaluating a Composite Function
Given \(f(x) = 2x + 1\) and \(g(x) = x^2 - 3\), find \(f(g(2))\) and \(g(f(2))\), and notice they aren't the same.
- For \(f(g(2))\), work from the inside out: first find \(g(2) = (2)^2 - 3 = 1\).
- Now substitute that result into \(f\): \(f(1) = 2(1)+1 = 3\). So \(f(g(2)) = 3\).
- For \(g(f(2))\), again start with the inner function: \(f(2) = 2(2)+1 = 5\).
- Substitute that into \(g\): \(g(5) = (5)^2 - 3 = 22\). So \(g(f(2)) = 22\).
- Since \(3 \neq 22\), \(f(g(x))\) and \(g(f(x))\) are different functions in general — order matters.
This is a common SAT trap: a question will define both \(f\) and \(g\) and ask for the two composites in different orders specifically to see whether you evaluate from the inside out correctly each time, rather than assuming composition is symmetric like multiplication.
Composite Functions in Context
Composite functions, written \(f(g(x))\), show up in word problems disguised as multi-step processes — one function converts units, and a second function uses that converted value. The rule is always the same: evaluate the innermost function first.
For example, if \(g(x)\) converts hours worked into dollars earned, and \(f(x)\) applies a tax deduction to dollars earned, then \(f(g(8))\) means: first find dollars earned for 8 hours using \(g\), then apply the tax deduction to that result using \(f\). Work from the inside out, always.
Where Transformations Show Up Elsewhere in Advanced Math
Transformations frequently combine with the exponential and quadratic functions covered in our exponential growth and decay guide and our nonlinear functions guide — a question might ask how shifting an exponential function up by \(k\) changes its horizontal asymptote, for instance. Building fluency with the four core transformations makes every one of those questions faster.
To build this fluency, practice with SAT practice questions that isolate function notation specifically — our Nonlinear Functions sets include this skill across all three difficulty tiers.
Frequently asked questions
Why does \(f(x-h)\) shift right instead of left?
Because the function needs a larger input to reach the same point on the curve — plugging in \(x=h\) into \(f(x-h)\) gives \(f(0)\), the same output the original function gave at \(x=0\), just shifted to a new location.
Do I need to know inverse functions for the SAT?
They appear far less often than shifts, stretches, and composite functions. Prioritize the four core transformations and composite function evaluation first.
What's the best way to avoid sign errors in combined transformations?
Break the transformation into separate pieces (shift, stretch, reflection) before combining them, exactly as shown in the worked example above, rather than trying to apply everything in one mental step.
Does \(f(g(x))\) ever equal \(g(f(x))\)?
Occasionally, for specific pairs of functions, but you should never assume it by default. Always evaluate each composite separately from the inside out, as shown above, rather than treating composition as interchangeable.