Composition conditions and a hidden constant case

Functions f:x↦(x2−6x−6)/(x+1) f:x\mapsto(x^2-6x-6)/(x+1), x∈Rx\in\mathbb R, x≠−1 x\ne-1; g:x↦(ax+1)/(x+b) g:x\mapsto(ax+1)/(x+b), x∈Rx\in\mathbb R, x≠−bx\ne-b, where a,ba, b are constants.

(i) Sketch y=f(x) y=f(x), giving coordinates of turning points and equations of asymptotes. Write down the range of ff. marks

(ii) Find the value of aa and range of bb such that both fgfg and gfgf exist. marks

(iii) Find g−1(x) g^{-1}(x). Given g−1(x)=g(x) g^{-1}(x)=g(x) for all real xx, x≠−bx\ne-b, find bb in terms of aa. marks

[2019/TJC/Promo/9]

Total marks: 10

Solution

(i)

Sketching rational functions

f(x)=x2−6x−6x+1f(x)=\dfrac{x^2-6x-6}{x+1} for x≠−1x\ne-1. Sketch y=f(x)y=f(x) giving the coordinates of the turning points and the equations of asymptotes. Write down the range of ff.

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Hint

  1. Divide to get f(x)=x−7+1x+1f(x)=x-7+\frac1{x+1}.

  2. Read off the asymptotes.

  3. Differentiate for the turning points.

Rational graph and its two branches
Rational graph and its two branchesScroll across to see the full diagram.

Answer · (i)

Asymptotes x=−1,y=x−7x=-1,y=x-7; maximum (−2,−10)(-2,-10), minimum (0,−6)(0,-6); range (−∞,−10]∪[−6,∞)(-\infty,-10]\cup[-6,\infty).

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Open interactive exploration

Two branches with an oblique asymptote

Switch on the asymptotes and the turning points.

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

  • F
  • Asymptotes
Ideas to try
Is any yy between −10-10 and −6-6 ever reached?