Question

Determine the domain of the function (f o g)((x) where:

Answer 1

Answer: You have the correct answer. It's choice B

Domain = $\left(-\infty, \frac{2}{5}\right)$

Nice work.

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Explanation:

The domain of g(x) is found by setting 2-5x greater than or equal to 0 and solving for x. We're doing this to ensure that 2-5x is not negative.

$2-5x \ge 0\\\\2 \ge 5x\\\\5x \le 2\\\\x \le \frac{2}{5}$

So we can plug in any number smaller than 2/5, or we can plug in 2/5 itself, into the g(x) function to get some output.

However, notice that if x = 2/5, then g(x) = 0. This then would feed into the f(x) function and lead to a division by zero error. Therefore, x = 2/5 must be kicked out of the domain of (f o g)(x). We keep everything else that we found earlier.

In short, the domain as an inequality is $x < \frac{2}{5}$, which is the same as saying $-\infty < x < \frac{2}{5}$ and that converts to the interval notation $\left(-\infty, \frac{2}{5}\right)$

We don't use a square bracket because we don't want to include the endpoint 2/5.