Question

The range of the function $\left.f(x)=\tan \sqrt{(} \pi^{\wedge} 2 / 9\right)-x^{\wedge} 2$ is
1) $[0,3]$
2) $[0,\sqrt{3}]$
3)$(-\infty ,\infty )$
4)None of these

Answer 1

Hint: A function's domain and range are its constituent parts. A function's range is its potential output, whereas its domain is the set of all possible input values.
"All the values" that are input into a function are referred to as the domain of a function. The collection of all potential inputs for a function is its domain.

Formula Used:
Finding the range of a function $y=f(x)$ is the same as finding all values that $y$ could be. The range of $f(x)$ is all the $y$-values where there is a number $x$ with $y=f(x)$.

Complete step by step Solution:
Given that
$\left.f(x)=\tan \sqrt{(} \pi^{\wedge} 2 / 9\right)-x^{\wedge} 2$
In this
$\left(\pi^{2} / 9\right)-x^{2} \geq 0$
$x^{2}-\left(\pi^{2} / 9\right) \leq 0$
Cross multiply the terms
$(x-[\pi / 9])(x+[\pi / 9]) \leq 0$
The group of inputs that make up a function's domain. The range is the collection of a function's outputs. When specifying domain and range, for example, interval notation is used to represent groups of numbers. There are either open or closed intervals or both.
Then we get the interval as
$x \in[-\pi / 3, \pi / 3]$
$-\pi / 3 \leq x \leq \pi / 3$
Which implies
$0 \leq x^{2} \leq \pi^{2} / 9$
$0 \geq-x^{2} \geq-\pi^{2} / 9$
Add $\text { Adding } \pi^{2} / 9$
$\pi^{2} / 9 \geq \pi^{2} / 9-\left[x^{2}\right] \geq 0$
\[\pi /3\ge {{\sqrt{\pi }}^{2}}/9-\left[ {{x}^{2}} \right]\ge 0\]
Taking tangent on both sides
\[\tan \pi /3\ge \tan {{\sqrt{\pi }}^{2}}/9-\left[ {{x}^{2}} \right]\ge \tan 0\]
$\sqrt{3} \geq \tan \sqrt{\pi} 2 / 9-\left[x^{2}\right] \geq 0$
The range is $[0, \sqrt{3}]$.

Hence, the correct option is 2.

Note: The collection of all a function's outputs constitutes its range. Example: Let's have a look at the function f: AB, where $f(x) = 2x$ and A and B each represent a "collection of natural numbers." The domain in this instance is A, and the co-domain is B. The range then appears as the function's output. The range is made up of even natural numbers. The elements of the co-domain that are mapped are known as the pictures, while the elements of the domain are known as pre-images. The set of all images of the domain's elements in this case serves as the function's range, as does the set of all of its outputs.