Composite and inverse functions – mapping diagrams, graphs and reflection in y = x

What it shows

Two functions f and g are built from five families: linear a·x + b, quadratic a·x² + b, exponential a·eˣ + b, square root a·√x + b and reciprocal a/x + b. The composite (f∘g)(x) = f(g(x)) is shown as a chained mapping diagram and as a graph with a construction through the line y = x. Its domain is the set of x in the domain of g with g(x) in the domain of f; domains and ranges are computed exactly, with endpoints rounded to 2 decimal places. The inverse is the reflection of the graph in y = x and exists only for one-to-one functions.

How to use

Start on Composite: set f and g with the a and b sliders, then drag x₀ on the graph and follow the arrows in the mapping diagram. Switch Order to compare f∘g with g∘f. On Sum & product, choose an Operation and watch the ordinate bars. On Inverse, pick a quadratic, tick Horizontal line test, then set Domain of f to x ≥ 0 and drag Point A to compare slopes. Reset restores the starting values.

Parameters you can change

• Screen Composite, Sum & product, Inverse
• Type of f Linear a·x + b, Quadratic a·x² + b, Exponential a·eˣ + b, Square root a·√x + b, Reciprocal a/x + b
• Coefficient a of f -5–5
• Coefficient b of f -5–5
• Type of g Linear a·x + b, Quadratic a·x² + b, Exponential a·eˣ + b, Square root a·√x + b, Reciprocal a/x + b
• Coefficient a of g -5–5
• Coefficient b of g -5–5
• Value x₀ (point A on the Inverse screen) -5–5
• Order of composition f∘g: f(g(x)), g∘f: g(f(x))
• Operation (Sum & product screen) f + g, f − g, f · g, f ÷ g
• Restrict the domain of f (Inverse screen) Whole domain, x ≥ 0, x ≤ 0
• Horizontal line test
• Horizontal line y = k -6–6

Questions to explore

1. For f(x) = √x and g(x) = x − 2, how do the domains of f∘g and g∘f differ?
2. Why does y = x² have no inverse on all real numbers, and how does restricting the domain help?
3. If the tangent to f at A has gradient 2, what is the gradient of the tangent to f⁻¹ at the mirror point?