lesson
Updated 1 month ago · 7 views
Imagine a relay race where runner A hands a baton to runner B.
In a composite function like (f∘g)(x), or f(g(x)), the inner function g finishes first, handing its output directly to the outer function f as its new input.
The Two-Step Visual Relay
To read f(g(x)) from a graph, you always work from the inside out in two distinct visual steps.
First, locate the given x on the graph of g to find its vertical y-value. Then, take that resulting number and find it on the horizontal x-axis of f to get your final y-value.
📊Interactive diagram
What happens when both functions are plotted on the very same coordinate plane? Let's trace through a complete example.
Worked Example: Evaluating f(g(2))
Let's evaluate f(g(2)) using two graphed curves: a line g(x) in blue and a parabola f(x) in purple.
📊Interactive diagram
Step 1: On the blue line g(x), find x=2. The graph shows (2,−1), which means g(2)=−1.
Step 2: Replace g(2) with −1 inside the outer function to get f(−1). Looking at the purple parabola f(x) at x=−1, the point is (−1,−4). Therefore, f(g(2))=−4.
What if we flip the order and evaluate g(f(2)) instead?