Question

How do you find the domain and range of \[g(x) = \dfrac{{2x - 3}}{{6x - 12}}\] ?

Answer 1

Hint: The domain of a function is the complete step of possible values of the independent variable. That is the domain is the set of all possible ‘x’ values which will make the function ‘work’ and will give the output of ‘y’ as a real number. The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain.

Complete step by step answer:
Given, \[g(x) = \dfrac{{2x - 3}}{{6x - 12}}\]. It can be simplified as, \[g(x) = \dfrac{{2x - 3}}{{6(x - 2)}}\]. We can see that \[g(x)\] is well defined at all the values of ‘x’ except at \[x = 2\].
That is for \[x = 2\] in \[g(x) = \dfrac{{2x - 3}}{{6(x - 2)}}\]
\[g(2) = \dfrac{{2(2) - 3}}{{6(2 - 2)}} \\
\Rightarrow g(2) = \dfrac{{4 - 3}}{{6(0)}} \\
\Rightarrow g(2) = \dfrac{1}{0} \\ \]
That is undefined value.Hence the domain is all the real numbers except at \[x = 2\]. Thus the domain is \[\{ x \in R:x \ne 2\} \].

Now to find the range We have \[g(x) = \dfrac{{2x - 3}}{{6x - 12}}\] or \[y = \dfrac{{2x - 3}}{{6x - 12}}\].
\[y = \dfrac{{2x - 3}}{{6x - 12}}\]
Cross multiplying we have,
\[y\left( {6x - 12} \right) = 2x - 3 \\
\Rightarrow 6xy - 12y = 2x - 3 \\ \]
Now grouping ‘x’ term in one side of the equation,
\[6xy - 2x = 12y - 3\]
Taking ‘x’ common in the left hand side of the equation,
\[x(6y - 2) = 12y - 3\]
Divide the whole equation by \[(6y - 2)\] on both side we have,
\[ \Rightarrow x = \dfrac{{12y - 3}}{{6y - 2}}\]

This can be simplified as
\[ \Rightarrow x = \dfrac{{12y - 3}}{{2(3y - 1)}}\]
we can see that ‘x’ is well defined for all real numbers except at \[3y - 1 = 0\].
That is at \[y = \dfrac{1}{3}\].
Now put \[y = \dfrac{1}{3}\] in \[x = \dfrac{{12y - 3}}{{2(3y - 1)}}\],
\[x = \dfrac{{12\left( {\dfrac{1}{3}} \right) - 3}}{{2\left( {3\left( {\dfrac{1}{3}} \right) - 1} \right)}} \\
\Rightarrow x = \dfrac{{4 - 3}}{{2\left( {1 - 1} \right)}} \\
\therefore x = \dfrac{1}{0} \\ \]
Which is undefined value.

Hence the range is \[\{ y \in R:y \ne \dfrac{1}{3}\} \] and the domain is \[\{ x \in R:x \ne 2\} \].

Note: When finding the domain remember that the denominator of a fraction cannot be zero and the number under a square root sign must be positive in this section. We generally use graphs to find the domain and range. But it is a little bit difficult to draw.