Question

Find the domain and range, $f(x) = \dfrac{{4 - x}}{{x - 4}}$

Answer 1

Domain of a function $f$on $x$ is the set of all possible values $x$can take and range is the set of all possible values $f(x)$ can take. Here the given function is a fraction. The denominator cannot be zero. So taking account of these things we can find domain and range.

Complete step-by-step answer:
Given the function, $f(x) = \dfrac{{4 - x}}{{x - 4}}$
It is a fraction. We know the denominator cannot be zero since division by zero is not defined.
So we have, $x - 4 \ne 0$
This gives, $x \ne 4$.
Therefore for the function, $x$ cannot take the value $4$.
And we can see that $x$ can take all other real numbers except $4$.
Domain of a function $f$on $x$ is the set of all possible values $x$can take.
So, the domain of this function is set of all real numbers except $4$ or simply $\mathbb{R} - \\{ 4\\}$.
Now let us check for the range.
Range is the set of all values $f(x)$ can take.
Here, $f(x) = \dfrac{{4 - x}}{{x - 4}}$
We can write it as,
$f(x) = \dfrac{{ - (x - 4)}}{{x - 4}}$
$\Rightarrow f(x) = - 1$
This gives for any value of $x$, $f(x)$ will be equal to $- 1$. So the only possible value of $f(x)$ is $- 1$.
So, the range of the function is $\\{ - 1\\}$.
Therefore, for the given function, $f(x) = \dfrac{{4 - x}}{{x - 4}}$

Domain is $\mathbb{R} - \\{ 4\\}$ and range is $- 1$.

Note: To define a function, we usually specify its domain and range. If not defined we can assume it as the set of real numbers. Here we could find the range easily by slight rearrangement. So we have to study the given function for finding domain and range.
For a function $f:A \to B$, $A$ is called the domain of the function and $B$ is called the range of the function.