Question

How do you find the domain and range of $\dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}?$

Answer 1

Let us first denote $f\left( x \right) = \dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}$ whose domain and range has to be determined. So first we need to know the definitions of domain and range of a function. Domain of a function refers to the set of possible values of x for which the function will be defined and the range refers to the possible values that the function can attain for those values of x which are in the domain of $f(x)$.
So the above problem is purely based on understanding the definitions of domain and range of a function and then finding each of them.

Complete step by step solution:
In the above problem given the function, $f\left( x \right) = \dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}$
We need to find the domain and range of $f(x)$ using the definitions of domain and range of a function which is mentioned in the hint already.
Firstly to find the domain we need to find the all possible values of x for which the function is defined. So, to do this we have to look at where the function goes undefined.
We make denominator equals to zero and proceed further.
Equating denominator to zero we get,
$\sqrt[4]{{{x^2} - 5x}} = 0$
Now raise both sides to the power 4 we get,
$\Rightarrow {\left( {\sqrt[4]{{{x^2} - 5x}}} \right)^4} = {0^4}$
Now cancelling fourth root on both sides, we get,
$\Rightarrow {x^2} - 5x = 0$
$\Rightarrow x(x - 5) = 0$
Hence we get the solution as $x = 0$ and $x - 5 = 0$
$\Rightarrow x = 0$ and $x = 5$.
The other possibility that might make the function undefined if the inside of the fourth root is negative.
To find when this happens, let us consider the inequality given below.
${x^2} - 5x < 0$
$\Rightarrow x(x - 5) < 0$
To find the negative possibilities for this inequality, we need to look at the intervals between the zeroes because that is where the function could go negative.
So for the function to be negative, only one of the products may be negative.
On the interval $( - \infty ,0)$ both factors will be negative, so their product will be positive.
On the interval $(0,5)$ one of the factors will be negative, hence we get a positive product.
On the interval $(5,\infty )$ both the factors are positive, so their product is also positive.
This gives us that the function is undefined only on the interval $(0,5)$.
Combining this with the zeroes from the denominator we get an undefined interval $[0,5]$.
Hence our domain of the function $f\left( x \right) = \dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}$ is $\\{ x|x < 0,x > 5\\}$.
Now we find the range of the function.
Note that the vertical asymptotes caused by the zeroes in the denominator would normally go to infinity from the positive direction and minus infinity from the negative direction.
Hence, we see that the root causes the negative values to be undefined. So we only get values greater than 0 to infinity. So the range of the function is all the positive real numbers.
Hence the range of the function $f\left( x \right) = \dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}$ is ${\mathbb{R}^ + }$.

Hence the domain and range of $f\left( x \right) = \dfrac{1}{{\sqrt[4]{{{x^2} - 5x}}}}$ is $\\{ x|x < 0,x > 5\\}$and ${\mathbb{R}^ + }$.

Note:
We must know the definitions of domain and range of the function to solve such problems. The above problem can +also be solved by the method of graphing, where we draw the graph and we find the values of domain and range from the values of x-axis and y-axis. This type of method is always solved by the concept of inequality. Solutions of inequalities are always written in the form of intervals.