f(a) is a height; a composite reads twice: To read f(a) off a graph: go up from x = a to the curve, then across to the y-axis. For (f∘f)(a) do it twice — read f(a) first, then feed that answer back in and read f of it.
IB-style question — read off the graph
The graph of y = f(x) for 0 ≤ x ≤ 8 is shown; it cuts the axes at (0, 6) and (8, 0). From the graph, f(2) = 3 and f(4) = 2. Find (a) f(4); (b) (f∘f)(4).
Step by step
- (a) Read the height of the curve at x = 4.
- (b) Composite — inside first: f(4) = 2, then read f(2).
Final answer
(a) f(4) = 2. (b) (f∘f)(4) = 3.
Use the answer as the next input: (f∘f)(a) is not f(a) doubled. Read f(a), then go to that x-value and read the curve again — keep the two read-offs separate.
f⁻¹ reverses the read — and reflects in y = x: To read f⁻¹(b) off the graph of f: start at y = b on the y-axis, go across to the curve, then down to the x-axis — that x is f⁻¹(b). To sketch f⁻¹, reflect the whole graph in the line y = x (every point (a, b) becomes (b, a), so the intercepts swap).
IB-style question — the inverse from the graph
Using the same graph (cutting the axes at (0, 6) and (8, 0), with f(2) = 3): (a) find f⁻¹(3); (b) state where y = f⁻¹(x) cuts the axes.
Step by step
- (a) f⁻¹(3) asks 'which input gives output 3?' Across from y = 3, down to the axis.
- (b) Reflecting in y = x swaps each intercept's coordinates.
Final answer
(a) f⁻¹(3) = 2. (b) f⁻¹ cuts the axes at (6, 0) and (0, 8).
[Diagram: math-inverse-reflection] - Available in full study mode
Intercepts swap: A y-intercept (0, k) of f becomes an x-intercept (k, 0) of f⁻¹, and vice versa. Mark the reflected intercepts when you sketch.