Question

How do you find the domain and range of \[\log \left( x-9 \right)\]?

Answer 1

Hint: This type of question is based on the concept of domains and ranges of a function. We have to first understand the given function. Since logarithm is not defined for 0 and negative terms, equate x-9>0. We get x>9. When x is 9 or less than 9, we get not defined functions. Hence, the domain is obtained. Substitute y=log(x-9). We find that by decreasing the value of x from infinity to $x >9$, the value of y becomes more and more negative. Hence, the range of the function y is also obtained.

Complete step-by-step solution:
According to the question, we are asked to find the domain and range of \[\log \left( x-9 \right)\].
We have been given the function \[\log \left( x-9 \right)\]. ----------(1)
Let us assume \[y=\log \left( x-9 \right)\].
We first have to find the domain.
Domain of this function is the set of values of x in which the function is defined.
Here, we have been given a logarithmic function.
We know that logarithm is not defined for 0 and negative terms.
Therefore, x-9 should be greater than zero, that is
x-9>0
Add 9 on both sides of the inequality. We get
x-9+9>0+9
On further simplification, we get
x>9
Therefore, the function is defined only for x>9.
Hence, the domain of the function y=log(x-9) is \[\left( 9,\infty \right)\].
Now, we have to find the range.
Range of this function is the set of all possible values of y.
We know that the domain of y=log(x-9) is \[\left( 9,\infty \right)\].
On substituting the values of x in the y, we find that the value of y becomes more negative as the value of x decreases from infinity to x>9.
In other ways, the value of y becomes more positive when x increases to infinity.
Hence, the range of y=log(x-9) is \[\left( -\infty ,\infty \right)\].
Therefore, the domain and range of \[\log \left( x-9 \right)\] is \[\log \left( x-9 \right)\] and \[\left( -\infty ,\infty \right)\] respectively.

Note: We should not put closed brackets instead of open brackets. Closed bracket is used only when the upper limit and the lower limit is considered. Here, for domain 9 is not considered, thus we use an open bracket. Don’t get confused with domain, range and codomain.