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

(i)

Range from a graph

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

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Hint

  1. Square-root graph starting where the radicand is zero.

  2. Read the range off the yy-values of the graph.

y=f(x)=√(x−2)
y=f(x)=√(x−2)Scroll across to see the full diagram.

Answer · (i)

Rf=[0,∞)R_f=[0,\infty); closed start (2,0)(2,0).

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The square root starts at (2, 0)

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

  • Y = √(x − 2)
  • Lower bound of the range
Ideas to try
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