Free practice

Function Practice Problems

Notation, piecewise, composition, and rate of change — with solutions.

Calculus is a course about functions, so fluency with functions themselves is the cheapest head start there is. Four ideas do most of the work, and each one turns into a calculus concept within weeks:

The function concept. One input, one output — checkable from a set of pairs, a table, or a graph. Precise reading of f(x) notation prevents an entire category of errors.
Piecewise functions. Knowing which branch owns the boundary point is exactly the skill behind one-sided limits and continuity questions.
Composition. Building f(g(x)) — and, harder, decomposing a given function into an outer and inner piece — is the chain rule's prerequisite. If you can spot the decomposition instantly, the chain rule is easy.
Average rate of change. The slope between two points on a curve is the derivative's “before” picture; comparing rates over different intervals from a table is a recurring AP question in its own right.

Round it out with domains in interval notation and graph transformations — shifting and stretching a known graph — and you have the function toolkit AP Calculus assumes on day one. The seven problems below cover each idea in order, with a worked solution after every answer.

Practice problems

Problem 1

Is the relation

{(3, 8), (- 4, 1), (3, - 7)}

a function?

Problem 2

f (x) = {x + 2 - x x < - 2 x \geq - 2

Find

f (- 2)

Problem 3

x - 4 - 2 - 1 1 f (x) 58 - 6 - 2 g (x) 6 - 2 - 2 - 1

Find

f (g (1))

Problem 4

h (x) = 8 x^{3} - 60 x^{2} + 150 x - 125

, which pair of functions satisfies

h (x) = f (g (x))

Problem 5

x f (x) - 2 - 1 - 1 - 2 318

Compare the average rate of change of

f

over the two intervals

Problem 6

A function

f

has domain equal to all real numbers except

x = - 2

and

x = 4

. Which interval notation describes this domain?

Problem 7

Let

f (x) = ∣ x ∣

. The point

(2, 2)

f

moves to

(1, - 1)

on the graph of

g

, a single horizontal and vertical translation of

f

. Which equation defines

g

Want more than 7 problems?

These are fixed examples. Lodera is an adaptive question bank for AP Calculus AB and Precalculus that never runs out: miss a question and it serves you variations of the same skill until it sticks. Every problem is generated from a template and solved by a computer algebra system — exact answers, no AI-written solutions. 10 free questions a day, forever.

Start practicing free