CAIE IGCSE Additional Math 1.5. Explain when inverses do not exist Question Bank

Practise deciding whether a function has an inverse and giving a clear reason based on whether one input can be recovered.

[Maximum number: 4]

The functions f and g are defined as follows, for all real values of x.

$\begin{aligned}f(x)&=2x^2-1\\g(x)&=e^x+1\end{aligned}$

For each of the functions f and g , either explain why the inverse function does not exist or find the inverse function, stating its domain.

f is not one-one, hence $f^{-1}$ does not exist oe.

B1.

$g^{-1}(x)=\ln(x-1)$
B2.

M1 for $x=\ln(y-1)$ and a swap of variables at some point; or $y=\ln(x+1)$ ; or $e^x=y-1$ and $y=\ln x-1$ .

Domain: x>1. B1.

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1.5. Explain when inverses do not exist question 1