When can two functions compose?

The functions are f(x)=x−2f(x)=\sqrt{x-2} for x≥2x\ge2, and g(x)=3x2−6x+5g(x)=3x^2-6x+5 for 0≤x≤40\le x\le4, with real inputs.

(i) Sketch y=f(x) and state its range. marks

(ii) Find the inverse of f and its domain. marks

(iii) Show that fg exists. marks

(iv) Find fg(x) and its range. marks

(v) With g now defined on all real numbers, describe fully a sequence of transformations taking y=g(x) to y=x2−1y=x^2-1. marks

[2023/MI/Promo/PU2/I/8]

Total marks: 12

Solution

If this part uses an earlier result, revisit (iii) Existence of composites or open all working together.

(iv)

Composite rule and range

Find fg(x)fg(x) and its range.

Explore this idea

Hint

  1. Substitute g(x)g(x) into ff.

  2. Simplify the radicand.

  3. For the range, use the range of gg as the new domain of ff.

Answer · (iv)

fg(x)=3∣x−1∣fg(x)=\sqrt3|x-1|, Rfg=[0,27]R_{fg}=[0,\sqrt{27}].

0≤x≤40\le x\le4.

Explore the idea

Open interactive exploration

A V with vertex (1, 0)

fg(x) = √3|x − 1| on the domain of g, 0 ≤ x ≤ 4.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Fg(x)
  • Maximum
Ideas to try
Why is the graph not symmetric about x=1x=1? (The domain stops at x=0x=0 on the left.)