Question

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

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

Answer:

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

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

Step-by-step 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 $\sqrt{x}\geq 0$ so  $-3\sqrt{x}\leq 0$, and thus $f(x)\in \mathbb{R}$, which leads us to affirm that the range of  $f$  is the set of all nonpositive numbers, $\{x\in \mathbb{R} : x \leq 0 \}$.