Largest possible domain and range
Largest possible domain and range
7 marks
TGM Original
Each function below is defined for the largest possible set of real values of \(x\).
\(\mathrm{f}(x)=\sqrt{6-2x}\qquad\mathrm{g}(x)=\ln(2x+5)\qquad\mathrm{h}(x)=\dfrac{3x+1}{x-2}\)
- Find the largest possible domain of each of \(\mathrm{f}\), \(\mathrm{g}\) and \(\mathrm{h}\).[3]
- Find the range of \(\mathrm{f}\).[1]
- By writing \(\mathrm{h}(x)\) in the form \(A+\dfrac{B}{x-2}\), find the range of \(\mathrm{h}\), explaining why one value is excluded.[3]
Solution:
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Answer:(a) \(\mathrm{f}\): \(x\le3\), \(\mathrm{g}\): \(x>-\dfrac52\), \(\mathrm{h}\): \(x\ne2\) (b) \(\mathrm{f}(x)\ge0\) (c) \(\mathrm{h}(x)=3+\dfrac{7}{x-2}\), range \(\mathrm{h}(x)\ne3\)
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