Solveeit Logo
Login

Question

Question: How do you find domain and range of \(f\left( x \right)=\dfrac{2x}{x-3}\) ?...

How do you find domain and range of f(x)=2xx−3f\left( x \right)=\dfrac{2x}{x-3} ?

Explanation

Solution

We have been given a fractional expression with linear functions in variable-x in both the numerator as well as the denominator. The denominator consists of a term in x-variable and a constant term, 3 but the numerator has no constant term. We shall find all values of variable-x for which the given function is defined to find the domain of the function and we shall find all the values of the function f(x) for every value of x given as input to it.

Complete step by step solution:
We can view a function, ff as something which takes input xx and for a given input, it produces an output which we call f(x)f\left( x \right).
Given that f(x)=2xx−3f\left( x \right)=\dfrac{2x}{x-3}.
Any fractional function has one condition for which it is defined. The denominator of the fraction must not be equal to zero.
Hence, x−3≠0x-3\ne 0.
We shall find the value of x for which the denominator of the fraction is zero, then we shall remove that value of x from the solution set of the domain of the function.
⇒x−3=0 ⇒x=3 \begin{aligned} & \Rightarrow x-3=0 \\\ & \Rightarrow x=3 \\\ \end{aligned}
Thus, the domain of the function is x\in \mathbb{R}-\left\\{ 3 \right\\}.
We shall find the inverse of the given function to find its range.
f(x)=2xx−3f\left( x \right)=\dfrac{2x}{x-3}
⇒y=2xx−3\Rightarrow y=\dfrac{2x}{x-3}
We shall first cross-multiply the terms and then write them in terms of variable-y.
⇒yx−3y=2x\Rightarrow yx-3y=2x
⇒yx−2x=3y\Rightarrow yx-2x=3y
⇒x(y−2)=3y\Rightarrow x\left( y-2 \right)=3y
⇒x=3yy−2\Rightarrow x=\dfrac{3y}{y-2}
⇒f−1(x)=3yy−2\Rightarrow {{f}^{-1}}\left( x \right)=\dfrac{3y}{y-2}
Therefore, y\in \mathbb{R}-\left\\{ 2 \right\\}.
Hence, the range of f(x)=2xx−3f\left( x \right)=\dfrac{2x}{x-3} is \mathbb{R}-\left\\{ 2 \right\\}.
Therefore, for function f(x)=2xx−3f\left( x \right)=\dfrac{2x}{x-3}, the domain is x\in \mathbb{R}-\left\\{ 3 \right\\}and the range is f\left( x \right)\in \mathbb{R}-\left\\{ 2 \right\\}

Note:
In the solution set of the range of a given function, we have excluded 2 from the set of all real numbers because the function is not defined for this value of variable-y. The way we had searched for values of x for which the function was not defined to remove them from the solution set while computing the range of the function, similarly we had removed 2 from the solution set of range of function.