Question

How do you find the domain and range of the function $f\left( x \right) = \log \left( {x - 2} \right)$?

Answer 1

In this question we are asked to find the domain and range of the function, this can be done by the definition of the domain and range of the function, The domain of a function $f\left( x \right)$ is the set of all values for which the function is defined, and the range of the function is the set of all values that $f$ takes, by using the definitions we will get the required result.

Complete step by step solution:
Given function $f\left( x \right) = \log \left( {x - 2} \right)$,
And by using the fact that the input values of the logarithm function should be greater than zero.
So, from the above we get,
$\Rightarrow x - 2 > 0$,
Now adding 2 to both sides we get,
$\Rightarrow x - 2 + 2 > 0 + 2$,
Now simplifying we get,
$\Rightarrow x > 2$,
So, the domain of the function is all the real numbers greater than 2,
Now for any logarithmic function $f\left( x \right)$comes out to be a real number, so the range of the function will be all real numbers.
So, the domain of the function is all real numbers greater than 2 which is given by interval notation$\left( {2,\infty } \right)$ and the range is all real numbers which is given by interval notation $\left( { - \infty ,\infty } \right)$.

**
$\therefore$ The domain of the given function $f\left( x \right) = \log \left( {x - 2} \right)$ is given by interval notation $\left( {2,\infty } \right)$ and the range which is given by interval notation $\left( { - \infty ,\infty } \right)$.**

Note:
A function is a relation between domain and range such that each value in the domain corresponds to only one value in the range. Relations that are not functions violate this definition. They feature at least one value in the domain that corresponds to two or more values in the range.