Question

Determine the domain of the function (f*g)(x) where f(x)=1/x, g(x)=√2-5x

Answer 1

Answer: (∞, 0) ∪ (0, 2/5)

In other words, x is less than 2/5 but cannot include 0

Explanation:

`f(x)\cdot g(x)=(1)/(x)\cdot(1)/(√(2-5x))\\\\\\\text{The denominator cannot equal zero and must be a real number, so}\\x\\eq 0\quad and\quad √(2-5x)>0\\\\\\(√(2-5x))^2>0^2\\.\quad 2-5x>0\\.\qquad -5x>-2\\\\.\qquad \quad x<(-2)/(-5)\\\\.\qquad \quad x<(2)/(5)\\\\\\\\\text{So, }x<(2)/(5)\quad \text{but cannot equal 0}\\\\\\\text{Interval Notation: }\large\boxed{ \bigg(-\infty,0\bigg) \text{U}\bigg(0, (2)/(5)\bigg)}`