# To find the domain of this function why don't we consider the numerator?

In the function f(x) = (4-x)/2+log base 2 of x
to find the domain we set the denominator different from zero and x>0 cause of the log but why dont we set 4-x different from zero? please help me clarify this doubt, thanks in advance

• Domain means where (= for which x) the function is defined (= where it can be computed). The fraction (4-x)/2 can be computed for any x, no problem when it's negative or zero. But the second tem uses the log function, which is only defined for positive arguments, but not when x <= 0. (Check the graph of log(x) - you can ask google "plot log(x)" : it goes to minus infinity when x approaches 0 from the right.) So we need x > 0. (The sum of the two terms is defined whenever both are defined.)

Let $f(x)=\frac{4-x}{2}+\log_2 ^{x}$.

Since $\log_2 ^{x}$ is not defined for $x\leq 0$, we must have $x>0$. On the other hand $\frac{4-x}{2}$ is defined for all $x$ (note that there is no division by zero). So the domain of $f$ is $x>0$ or
$(0,\infty).$

You do not set $4-x$ to be zero, because $4-x$ in not in the denominator. It is fine to devide 0 by any number, but we should avoid dividing by zero. In the expression $\frac{4-x}{2}$ you never divide by zero.

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