The domain is the set of possible values for x. Normally we assume that x can be any real number. But we have to ensure that the various taboos in Math are not violated. What taboos are there in Math? Here are a few of the most well-known:
Zeros in denominators. Dividing be zero is undefined.
Zero or negative bases or arguments of logarithms.
Negative radicands (the number inside a radical) of even-numbered roots (like square roots). We cannot allow something like to happen because it is not possible to raise any real number to an even power and get a negative result.
Your function has no denominators or even-numbered roots. But it does have a logarithm. We must make sure that its base and argument are always positive. The base is 3 which is positive. The argument is x-4. To make the argument positive we need:
or
This is our domain.
Vertical asymptote. For logarithmic functions the vertical asymptote will be at the "edge" of the domain. To find it, just make an equation out of the domain:
x = 4
Range. Range is the set of possible y values. In this function your y value is a logarithm. Logarithms are exponents and exponents can be any number. So the range is all real numbers.
X-intercept. The x-intercept(s) of a graph are the points where the graph crosses/touches/intersects the x-axis. Since all the points on the x-axis have y coordinates of 0, we find x-intercepts by setting y to zero and solving for x:
To solve this we rewrite it in exponential form:
which simplifies to:
Adding 4 to each side we get:
So there is one x-intercept: (5, 0)
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