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 (iv) Composite rule and range or open all working together.

(v)

Transformations

With gg now defined for all real xx, describe fully a sequence of transformations that takes the graph of y=g(x)y=g(x) to the graph of y=x2−1y=x^2-1.

Explore this idea

Hint

  1. Complete the square: g(x)=3(x−1)2+2g(x)=3(x-1)^2+2.

  2. Remove the xx-translation, then handle the stretch and the vertical shift.

  3. Track the vertex: (1,2)→(0,−1)(1,2)\to(0,-1).

Answer · (v)

Translate left 1, down 5, then scale vertically by 1/3.

Explore the idea

Open interactive exploration

From 3(x − 1)² + 2 to x² − 1

Translate left, translate down, then squash vertically.

Teal: y = 3(x − 1)² + 2. No comparison overlay.

Start: y = 3(x − 1)² + 2

Vertex (1,2)(1,2).

  • Graph at the selected step
  • Vertex
Ideas to try
Follow the vertex through the four stages.