Question

How do you find the domain and range of $y = \log (2x - 12)$ ?

Answer 1

Hint : A function is defined in an equation defined in terms of a variable, as the value of the variable changes the values of the function also changes. All the possible values that the variable can take is called the codomain of a function, the domain of the function is defined as the values of the variable for which a function is defined, and the range of the function is defined as the set of all the possible values that a function can attain. We can find out the domain and range of the given function by using the above-mentioned definition of domain and range of a function.

Complete step-by-step answer :
We are given that $y = \log (2x - 12)$
We know that the input values of the logarithm function should be greater than zero.
So,
$2x - 12 > 0 \\\ \Rightarrow 2x > 12 \\\ \Rightarrow x > 6 \;$
Thus, the domain of $f(x)$ is all real numbers greater than 6.
Now, for any logarithm function, $y = a{\log _c}(b(x - h)) + k$ the value of y comes out to be a real number, so the range of the function is all real numbers.
Hence, the domain of $y = \log (2x - 12)$ is all real numbers greater than 6 and the range of $y = \log (2x - 12)$ is all the real numbers.

Note : For solving this kind of question, we must know the concept of the domain and range of function clearly. We know that the logarithm functions are the inverse of exponential functions that’s why the argument of the logarithm function is taken as greater than zero, that is, it cannot be negative or zero.