1. what are the domain and range of the function? f(x) = \sqrt[3]{x - 3} 2.what are the domain and range of the function? f(x) = - 3 √(x) ​

Question

1. what are the domain and range of the function?


`f(x) = \sqrt[3]{x - 3}`
2.what are the domain and range of the function?

`f(x) = - 3 √(x)`
​

Answer 1

1) domain of the first is ( R ) because of its root(3) and its range is (R) because of its root again

2)and domain of the last , is [0,+infinity)

and for its range , you can draw it such as picture and will be [0,- infinity)

Answer 2

1.
`Dom(f)=\mathbb{R}`,
`Ran(f)=\mathbb{R}`.

2.
`Dom(f)=\{x\in \mathbb{R} : x\geq 0 \}`,
`Ran(f)=\{x\in \mathbb{R} : x \leq 0 \}`.

Explanation:

1. Since the degree of the radical is an odd number, the radicand can be any real number, then
`x` can take any real value, so the domain of
`f` is the set of all real numbers,
`\mathbb{R}`.

Now, if
`x\in \mathbb{R}` , then
`x-3 \in \mathbb{R}`, so
`\sqrt[3]{x-3} \in \mathbb{R}`, and thus
`f(x)\in \mathbb{R}`, which leads us to affirm that the range of
`f` is the set of all real numbers,
`\mathbb{R}`.

2. Since the degree of the radical is an even number, the radicand can not be a negative number, then
`x` can take only nonnegaive values, so the domain of
`f` is the set of all nonnegative numbers,
`\{x\in \mathbb{R} : x\geq 0 \}`.

Now, if
`x\geq 0` , then
`√(x)\geq 0` so
`-3√(x)\leq 0`, and thus
`f(x)\in \mathbb{R}`, which leads us to affirm that the range of
`f` is the set of all nonpositive numbers,
`\{x\in \mathbb{R} : x \leq 0 \}`.