Zero and Divisions. In f(x) = (27x-(3^x)x) / ((3^x)x-9x)f(x)=27x−(3x)x(3x)x−9x find the Zero, Undefined, and Indeterminate. Please teach me how?

Answers are:
Zero = 3
Undefined = 2
Indeterminate = 0
I just don't know how to get them.

1 Answer
Jake M.
Jul 19, 2018

For each of these, we need to think about where the zeroes for the numerator and denominator are. Let's explicitly write that

n(x) = (27-3^x)x and d(x) = (3^x-9)x n(x)=(27−3x)xandd(x)=(3x−9)x

Clearly, n(x) = 0n(x)=0 when x = 0 or 3x=0or3 and d(x) = 0d(x)=0 when x = 0, 2x=0,2.

A zero happens when n(x)n(x) is zero and d(x)d(x) is some number, so we get 0/text(some number) = 0 0some number=0.

An undefined value happens whens d(x)d(x) is zero and n(x)n(x) is some number, so we get text(some number)/0some number0 which is undefined.

An indeterminate case is if both d(x)d(x) and n(x)n(x) are zero, since 0/000 is indeterminate.

Thinking about all three of these cases, we can easily derive the solution you gave: zero at x=3x=3, undefined at x=2x=2 and indeterminate at x=0x=0.

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