A problem related to domain and range of real functions.

Question

I am trying to solve a problem related to real functions. For which I need properties of domain and range which a number must follow to be in domain and range of the real function.

I have found the property, for domain but I'm not able to find of range.

Since functions can have different domains and range, hence they can have different properties of domain and range. So let's consider a function $f$. Which is defined as:

\begin{equation} f=\{(x, y): \quad y=\sqrt{16-x^{2}} \ and\ x, y \in R\} \end{equation}

Now property of Domain for this function is:

For any real number $a$ to be in domain of function $f$ there must exist only one real number $b$ such that:

\begin{equation} b=\sqrt{16-a^{2}} \end{equation}

I am not able to construct similar statement for range of the function. So I need your help. So if you're going to answer this question then please keep it in mind that I am talking about "real functions" ie functions whose both domain and codomain are either subset of $R$ or are $R$. Thanks.

Answer 1

The range is characterized as the set of all real numbers $y$ such that $f(x)=y$ for at least one $x$ in the domain of $f$.

[The domain consists of all $x$ with $16-x^{2} \geq 0$ which means $-4 \leq x \leq 4$. The range consists of all non-negative real numbers less than or equal to $4$. [If $0 \leq y \leq 4$ then $x=\sqrt {16-y^{2}}$ satisfies $f(x)=y$].