Question

What is the domain and range of $\dfrac{x-1}{x-4}$?

Answer 1

Use the definition of domain and range of a function. For domain we find the values where the function is defined, so we deduct the value where the function is undefined. Equating the denominator to 0 we get those values of x for which y is undefined. For Range we convert the function in terms of y such that we find the values of y which x is defined using the same procedure.

Complete step by step solution:
Domain of a function $y=f\left( x \right)$is a set of all those values for which function is defined. So, domain contains all possible values of x for which y exists.
Range of a function contains all the possible values of y for which x exists.
Let us solve the given question,
Let us assume the given function is $y=\dfrac{x-1}{x-4}$………. (1)
Domain:
The domain of the function will include all possible values of x except the value that makes the denominator equal to zero.
A function will not be defined if its denominator is 0
Equating the denominator of the function to 0
$\Rightarrow x-4=0$
Shifting the constant values to right hand side of the equation
$\Rightarrow x=4$
So, all the values except 4 are contained in the domain of the function.
We can write the domain of $y=\dfrac{x-1}{x-4}$is $\left( -\infty ,-6 \right)\cup \left( -6,\infty \right)$.
Range:
Since we know the range consists of values of y for which x exists, we will convert the function in terms of x to the function to the function in terms of y.
We have the function $y=\dfrac{x-1}{x-4}$
Switching the variables from x to y and vice versa
$\Rightarrow x=\dfrac{y-1}{y-4}$
$\Rightarrow x=\dfrac{\left( y-4 \right)+3}{y-4}=1+\dfrac{3}{y-4}$
Since the fraction $\dfrac{3}{y-4}$can never be equal to zero, the function can never take the value
$\Rightarrow y=1+0=1$
The range of the function will be $\left( -\infty ,1 \right)\cup \left( 1,+\infty \right)$

Note: Whenever we face such questions the key concept is to be clear about the definitions of domain and range. Do not mistake the closed brackets for range and domain here, keep in mind we have just the value that occurs before that number where the function is undefined but we have to exclude the value where the function is undefined.