Functions

• 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)
• Finish on one ACT-hard question and say what made it hard

A function f(x) assigns exactly one output to each input. The notation 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. For the range, find the lowest and highest outputs: f(x) = (x − 3)² + 2 can never go below 2, so its range is y ≥ 2, and a graph's range runs from its lowest point to its highest point on the y-axis. To decide which equation fits a table of values, test every column in each choice, because a wrong equation often fits one or two. The average rate of change of f from x = a to x = b is (f(b) − f(a))/(b − a), the slope between those two points. 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. Functions come up regularly on the Math section. Pick a number also settles 'which expression is equivalent' questions about functions: evaluate the original at a simple input and compare each choice at the same input. The hardest questions, at the end of the section: questions that run a function backward or move a point. If f(x) = 3x − 2 and f(g(x)) = 6x + 7, then 3g(x) − 2 = 6x + 7, so g(x) = 2x + 3; if (2, 5) is on the graph of y = f(x), then (−2, 14) is on y = 3f(x + 4) − 1, because x + 4 = 2 gives x = −2 and 3(5) − 1 = 14.

• 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.