Question

How do you find the domain of \[f\left( x \right)=\dfrac{1}{x-3}\] ?

Answer 1

Hint: These types of problems are pretty straight forward and are very easy to solve. This problem is a combination of algebra and functions which are used to determine the domain of various functions. For finding out the domain of any function, we need to first of all eliminate all the points or all the range of points of the function, corresponding to which the function becomes undefined. We can do this by performing a simple analysis. The rest of the points left, constitute the domain of the function.

Complete step-by-step solution:
Thus we start off with the given problem, we first try to eliminate all the points or range of points which don't satisfy the function or due to which the function becomes undefined. Analysing the problem closely, we see that if we put \[x=3\] or we find the value of \[f\left( 3 \right)\] , then it comes out to be infinity. This line \[x=3\] is thus an asymptote to the graph \[f\left( x \right)=\dfrac{1}{x-3}\] , as it is the line that meets or intersects the given curve at infinity. We eliminate such points from our domain. Now, after we have eliminated the point \[x=3\] we can observe very clearly that for any other value of \[x\], the value of the function is defined, hence we can safely say that the domain of the function or the possible values of \[x\] for which the function is defined and has a real value, is all real values except \[x=3\] . We can mathematically represent it as \[x\in \left( -\infty ,\infty \right)-\left\{ 3 \right\}\] or we write it as, \[x\in R-\left\{ 3 \right\}\]

Note: These types of problems can also be done by another method, which is basically the graphical method. In this method, we plot the function \[f\left( x \right)=\dfrac{1}{x-3}\] , and check for any points of discontinuity or any of the points which may lead to an undefined value or the asymptote lines. If we plot the graph we can clearly observe that function goes on from \[\left( -\infty ,\infty \right)\] except at the line \[x=3\] where it approaches infinity. This clearly explains that the domain of the function is \[x\in R-\left\{ 3 \right\}\] .