Let $a\gt1$ and $0 \lt b \lt 1$. If $f: \mathbf{R} \rightarrow[0,1]$ is defined by…
Let $a\gt1$ and $0 \lt b \lt 1$. If $f: \mathbf{R} \rightarrow[0,1]$ is defined by
$f(x)=\left\{\begin{array}{l}a^x,-\infty \lt x \lt 0 \\ b^x, 0 \leq x \lt \infty\end{array}\right.$ then $f(x)$ is
A bijection
One-one but not onto
Onto but not one-one-
Neither one-one nor onto
Solution
$f(x)= \begin{cases}a^x, & -\infty \lt x \lt 0 \\ b^x, & 0 \leq x \lt \infty\end{cases}$
$a\gt1$ and $0 \lt b \lt 1$
So by the graph
Clearly, every horizontal line cuts $f(x)$ at 2 points. Also, for $f(x)=0$
There is no $x \in \mathbb{R}$
So, $f(x)$ is neither one-one nor onto
