Originally Posted by
undefined OK, I see the worksheet itself is a little hard to understand the way it's written, but I'll try to answer:
a) $\displaystyle g(x)=x-1$ is a function for all real x. This could be written in several different ways, including $\displaystyle x\in\mathbb{R}$ and $\displaystyle x\in(-\infty, \infty)$.
b)
Greater than zero: $\displaystyle x\in(1, \infty)$
Less than zero: $\displaystyle x\in(-\infty, 1)$
Equal to zero: $\displaystyle x=1$
c) The domain of $\displaystyle f(x)$ is $\displaystyle x\in[1, \infty)$. This corresponds to $\displaystyle g(x)\geq0$.
Follow up question: $\displaystyle f(x)=\sqrt[3]{x-1}$ is a function for $\displaystyle x\in\mathbb{R}$ because all real numbers have exactly one real cube root. Thus as long as $\displaystyle x-1$ is a real number, $\displaystyle f(x)$ will associate it with a unique value. The range is also all real numbers.
Integrative Activity: For a function $\displaystyle f(x)=\sqrt{h(x)}$, the domain of $\displaystyle f(x)$ is described by $\displaystyle h(x)\geq0$.