Question

If $f:R^{+}\to R^{+}$ be defined as $f\left(x\right)=\{\begin{array}{c}{2}^x,x\in (0,1)\\ 3^x,x\in [1,\infty )\end{array},$
then which among the following options is correct?

• A. $f\left(x\right)$ is many one function
• B. $f\left(x\right)$ is one-one function
• C. Range of $f\left(x\right)$ is $(0,\infty )$
• D. Range of $f\left(x\right)$ is $(0,2)\cup (3,\infty )$

Answer 1

The correct option is B $f\left(x\right)$ is one-one function


From the above graph, using horizontal line test we can say that $f\left(x\right)$ is one-one function and it's range is $(1,2)\cup [3,\infty )$.

$\text{Alternate Solution}:$
$f\left(x\right)=\{\begin{array}{c}{2}^x,x\in (0,1)\\ 3^x,x\in [1,\infty )\end{array}$
Case$:1x\in (0,1)$
For $x_1,x_2\in (0,1)$
Let, $f\left(x_1\right)=f\left(x_2\right)$
$2^{x_1}=2^{x_2}$
$\Rightarrow x_1=x_2$
Case$:2x\in [1,\infty )$
For $x_1,x_2\in (0,1)$
Let, $f\left(x_1\right)=f\left(x_2\right)$
$3^{x_1}=3^{x_2}$
$\Rightarrow x_1=x_2$
In both the cases, for $f\left(x_1\right)=f\left(x_2\right)$ we get, $x_1=x_2$
So, $f\left(x\right)$ is one-one.