Function notation and composition
Function questions are written in a notation that hides how simple they are. The bracket is an instruction, not a multiplication, and once you read it that way most of these questions become substitution.
The bracket is an instruction
f(3) means put 3 wherever x appears in the rule and work it out. If f(x) = 5 - 2x, then f(3) is 5 - 6, which is -1. The notation misleads when the input is an expression: f(a + 1) means substitute a + 1 for every x, brackets included, so 5 - 2(a + 1) becomes 3 - 2a. Dropping the brackets around the substituted expression is the most common error in the topic.
Reading a value from a table or a graph
A point on the graph of f at (4, 7) says f(4) = 7, and nothing else. A table works the same way: find the input in the first column and read the output in the second. Questions that ask for f(4) while showing only a graph are not asking you to find a formula, so do not invent one. When the graph shows a line or a curve, the answer is the height at that x.
Composition works inside out
f(g(2)) means evaluate g at 2 first, then feed that result into f. Inside out, always. If g(x) = x + 1 and f(x) = 3x, then g(2) is 3 and f(3) is 9. Writing f(g(x)) as a single expression is a different job: substitute the whole rule for g into every x of f, which gives 3(x + 1). Both versions appear, and the order of operations is the same in each.
Inverses are a question, not a symbol
f⁻¹(18) means the input that produces 18, so solve f(x) = 18. With f(x) = 3x - 7, that is 3x - 7 = 18, so x = 25 and f⁻¹(18) = 25. To find the rule for an inverse, swap x and y and solve for y. The one thing the symbol never means is the reciprocal: f⁻¹ is not 1 over f.
Domain is a restriction, not a decoration
Two restrictions cover almost every domain question. A denominator cannot be zero, so any x that makes it zero is excluded. A square root needs a non-negative input, so the expression under the sign must be at least zero. For f(x) = 1 over (x - 4), the domain is every number except 4. For f(x) = the square root of x - 4, the domain is x at least 4.
Shifts from the notation
The position of the change tells you its direction. Adding inside the bracket moves the graph horizontally, and in the opposite direction to the sign: f(x - 2) shifts right by 2, and f(x + 2) shifts left by 2. Adding outside the bracket moves it vertically: f(x) + 3 shifts up by 3. A negative sign outside flips the graph over the x axis. Questions usually describe the movement and ask which expression matches, so read the direction from the notation rather than trying to picture the whole curve.
What does f(3) mean?
Substitute 3 for x in the rule for f, then simplify. If f(x) = 2x + 1, then f(3) is 7. The brackets never mean multiplication.
How do I evaluate f(g(2))?
Work from the inside out. Find g(2) first, then put that number into f. Reversing the order is the most common mistake in composition questions.
How do I find f inverse of 18?
Set the rule equal to 18 and solve for x. The answer is the input that produces 18, not the reciprocal of f.
What is the domain of a square root function?
Every x that makes the expression under the root at least zero. For the square root of x - 4, that is x at least 4.
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