Question

Which one of the following functions is not one-one?

• A. $f:(-1,\infty )\to R$ given by $f\left(x\right)=x^2+2x$
• B. $g:(2,\infty )\to R$ given by $g\left(x\right)=e^{x^3-3x+2}$
• C. $h:R\to R$ given by $h\left(x\right)=2^{x(x-1)}$
• D. $\varphi :(-\infty ,0)\to R$ given by $\varphi \left(x\right)=\frac{x^2}{x^2+1}$

Answer 1

Answer: (C)

$h:R\to R$ given by $h\left(x\right)=2^{x(x-1)}$

$x(x-1)$ is not a one -one function in whole $R$.

This function has a minima at $x=\frac{1}{2}$, therefore repeats value after $x=\frac{1}{2}$.

We can also see that $x(x-1)=0$ for $x=0,1$ and therefore $e^{x(x-1)}=1$ for both $x=0\&1$, which rules it out as a one to one function.