Functions

What you'll learn

• Evaluate f(x) for given values and expressions
• Determine domain and range from equations and graphs
• Perform function composition: (f ∘ g)(x) = f(g(x))
• Interpret function transformations (shifts, reflections, stretches)

Key concepts

A function f(x) assigns exactly one output to each input. F(3) means 'substitute 3 for x in the function rule.' The domain is all possible input values; the range is all possible output values. For f(x) = 1/(x − 2), the domain excludes x = 2 because division by zero is undefined. Composition f(g(x)) means evaluate g first, then use that result as input to f. Transformations: f(x) + k shifts up, f(x − h) shifts right, −f(x) reflects over the x-axis, and f(−x) reflects over the y-axis. The ACT tests functions on approximately 4-6 questions per exam.

Pro tips

• For f(g(x)), always work from the inside out. Evaluate g(x) first, then plug the result into f.
• If asked for the domain of a fraction, set the denominator ≠ 0; for a square root, set the expression inside ≥ 0.
• On graph-based function questions, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.