SAT Math — function notation: evaluating, composing, shifting and reflecting

The Nonlinear functions skill of the digital SAT's Advanced Math domain expects fluent use of function notation. Many questions are short, but each one tests whether you know exactly what the notation asks for: an input changed before the rule is applied, an output changed after it, or one function applied to the result of another.

This material starts with evaluation and composition: rewriting f(x + 2) as an expression, computing g(f(2)) from the inside out, and writing f(g(x)) as a single expression, with distractors built from the most common confusion — composing in the wrong order. A table of values is used for a double evaluation f(f(2)), and one question asks for the values of x that keep a composition with a square root defined.

The second part links notation to graphs. You find the image of a point under y = −f(x) + 3, the zeros of g(x) = f(x − 4), the vertex of the graph after f(x + 4) − 5, and an equivalent form of 2^(x + 1). A context question interprets h(3) = 53 in a height model with the right units, another solves f(a) = 12 under a condition, and the last finds a constant from one input-output pair before evaluating the function elsewhere. Throughout, the explanations separate what happens to the input from what happens to the output, the distinction behind most wrong answers on these questions.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

• Evaluate and simplify expressions such as f(x + 2) for a nonlinear function
• Compute and write compositions f(g(x)) and g(f(x)) and tell them apart
• Interpret f(a) = b in a real-world model with units
• Describe the effect of f(x − h), f(x) + k and −f(x) on points, zeros and vertices
• Find an unknown constant in a function rule from a known value

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: nonlinear functions — function notation, composition and transformations), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.