AI Tutor
Imagine scanning a flight schedule to find your layover city, then looking up that city on a second board to find your final gate number.
Evaluating composite functions from data tables uses this exact two-step lookup process.

The Inside-Out Rule

In a composite function (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)), you evaluate the inner function g(x)g(x) first. That resulting output becomes the input value for the outer function ff.
How do we trace this step-by-step when evaluating numerical expressions?

Step-by-Step Table Lookup

Let's evaluate (f∘g)(2)(f \circ g)(2) using tables for f(x)f(x) and g(x)g(x).
First, locate 22 in the input row of table gg to find g(2)=5g(2) = 5. Then locate 55 in the input row of table ff to get f(5)=−1f(5) = -1, giving (f∘g)(2)=−1(f \circ g)(2) = -1.
Will swapping the order of the functions yield that exact same result?

Order Matters

To evaluate (g∘f)(2)=g(f(2))(g \circ f)(2) = g(f(2)), you must start with the inner function ff instead of gg.