Nonlinear Functions and Function Notation
Function notation and exponentials ask what the symbols mean, not just how they move. The notation f(x) is an instruction to substitute an input into a rule, a composition runs from the inside out, and an exponential equation asks what power of the base reaches the target. The algebra is short. The points leak in four moves: f(x) read as multiplication, a composition run in the wrong order, an exponential linearized into base times exponent, and a growth model's initial value, factor, and output swapped.
These patterns aren't really about whether you can evaluate an expression. They're about whether the input landed inside the rule, whether the inner function ran first, whether the exponent stayed a power instead of becoming a coefficient, and whether you read each part of the model as the part it is.
The work
4 ways in · any order
Lesson
Nonlinear Functions and Function Notation
›
Substitute into f(x) as a rule, run compositions from the inside out, solve exponential equations by asking what power instead of dividing, and read a growth model part by part. The lesson works the method and the four patterns that derail it, and it saves the order trap for last: f(g(x)) computed as g(f(x)).
Diagnostic
10-item topic check
›
Ten items across the four patterns: reading f(x) as multiplication, composing in the wrong order, linearizing an exponential equation, and misreading a growth factor, rate, or initial value. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
›
About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
›
Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.