Answer: You have the correct answer. It's choice B
Domain = [tex]\left(-\infty, \frac{2}{5}\right)[/tex]
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.
[tex]2-5x \ge 0\\\\2 \ge 5x\\\\5x \le 2\\\\x \le \frac{2}{5}[/tex]
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 [tex]x < \frac{2}{5}[/tex], which is the same as saying [tex]-\infty < x < \frac{2}{5}[/tex] and that converts to the interval notation [tex]\left(-\infty, \frac{2}{5}\right)[/tex]
We don't use a square bracket because we don't want to include the endpoint 2/5.
