Question

Which statements describe the graph of y = Negative RootIndex 3 StartRoot x minus 1 EndRoot + 2? Select three options.

The graph has a domain of all real numbers.
The graph has a range of y greater-than-or-equal-to 1.
As x is increasing, y is decreasing.
The graph has a y-intercept at (0, 1).
The graph has an x-intercept at (–7, 0).

Answer 1

The graph has a domain of all real numbers.

The graph has a y-intercept at
`(0,1)`.

The graph has an x-intercept at
`(-7,0)`.

Explanation:

Given: The graph is
`y=\sqrt[3]{x-1}+2`

The domain of a function is a set of input values for which the function is real and defined.

Thus, the graph has a domain of
`(-\infty, \infty)`.

To find the y-intercept: To find the y-intercept, substitute
`x=0` in
`y=\sqrt[3]{x-1}+2`.

`\begin{aligned}y &=\sqrt[3]{x-1}+2 \\&=\sqrt[3]{0-1}+2 \\&=-1+2 \\&=1\end{aligned}`

Thus, the y-intercept is
`(0,1)`

To find the x-intercept: To find the x-intercept, substitute
`y=0` in
`y=\sqrt[3]{x-1}+2` .

`\begin{aligned}y &=\sqrt[3]{x-1}+2 \\0 &=\sqrt[3]{x-1}+2 \\-2 &=\sqrt[3]{x-1} \\(-2)^(3) &=(\sqrt[3]{x-1})^(3) \\-8 &=x-1 \\-7 &=x\end{aligned}`

Thus, the x-intercept is
`(-7,0)`