Question

Identify the function from the description given.

1. Its an irrational function

2. Its graph is mirror image of y =$x^{2n}$ about y = x, for some n $\ge$ 1, n $\in$ I and x $\ge$ 0

3. x $\le$ f(x) $\le$ $x^{1/6}$ if x $\in$ [0, 1]

4. Its domain and range are [0, $\infty$)

• A. $\frac{1}{x^{\frac{1}{8}}}$
• B. $\frac{1}{x^{\frac{1}{2}}}$
• C. $\frac{1}{x^{\frac{1}{2}}}$
• D. $x^{\frac{1}{8}}$

Answer 1

The correct option is C

$\frac{1}{x^{\frac{1}{2}}}$

We have to find the function which satisfies the 4 conditions from the options.

First condition is it's an irrational function.

All the functions given in the options are irrational functions.

Second condition says its graph is the mirror image of y = $x^{2n}$ about y = x. This means our function f(x) is the inverse of $x^{\frac{2}{n}}$ for some n we will find the inverse of $x^{2n}$.
y = $x^{2n}$

$\Rightarrow$ x = $y^{\frac{1}{2n}}$

$\Rightarrow$ f(x) = $x^{\frac{1}{2n}}$

It means our function will be of the form $x^{\frac{1}{2n}}$ , n $\ge$ and n $\in$ I.

Only $x^{\frac{1}{2}}$ and \ are of the form $x^{\frac{1}{2n}}$.

Next condition given is x $\le$ f(x) $\le$ $x^{\frac{1}{6}}$

$x\le x^{\frac{1}{2}}\le x^{\frac{1}{6}}$ , when x $\in$ [0,1] . $x^{\frac{1}{s}}$ does not satisfy this.

$\Rightarrow$ $x^{\frac{1}{2}}$ is our function.