Hi, everyone! Welcome back to integralcalc.com. Today we’re going to be doing another problem
with domain and range. And in this one, we’ve been asked to find the domain of the function
f(x) = 10^2x + log(21 – 3x). And what we need to remember about domain is that the
domain of a function is all of the numbers for which the function is defined. The easiest
way to figure out domain is by first finding where the function is not defined, if anywhere.
And then just exclude those values from your domain. So you need to look for the give-aways.
In this case, we’ve kind of got two terms here. We have 10^2x and we have log (21 – 3x)
and we can treat those separately. There’s nothing really restrictive about
10^2x. There’s no number that you could plug in for x, if you’re just looking at
10^2x, that would make this part of this function undefined. So there aren’t really any values
there that we have to consider not being in the domain. But it’s a different story with
the log function here because as you may or may not know, you can not take log of zero
or of a negative number. What that means is that 21 – 3x must be greater than zero.
Everything inside the log function must be greater than zero because again, log of zero
is undefined and log of a negative number is undefined. We can solve this inequality
by adding 3x to both sides and we get 21 > 3x. If we divide both sides by 3, we’ll get
that 7 must be greater than x or in other words, that x has to be less than 7. So in
order for this function to be defined, x has to be less than 7 which means that the domain
we can just write as x must be less than 7. Not < 7 but < 7. If the domain were 7 itself,
we would get zero inside the log function and that’s undefined. Remember that domain
is all values of x where the function is defined and in this case, that is all values less
than 7. So the domain is that x has to be less than 7. Anyway, that’s it. I hope that
video helped you guys and I will see you in the next one. Bye!
