We say a function $y=f(x)$ is defined for $x=a$ when $f(a)$ is a real number. [We will talk about (6)–(11) in more detail in Chapter 3. If you have no prior knowledge about them, you may skip them.]

1.$f(x)=x^{m}$ where $m$ is any positive integer is defined for every value of $x$.

2. $f(x)=\dfrac{1}{x}$ is defined for every value of $x$ except $x=0$, since $\frac{1}{0}$ cannot be calculated.

3.$f(x)=\sqrt{x}$ is defined for $x\geq0$ [because for $x<0$, $\sqrt{x}$ is imaginary].

4. $f(x)=\sqrt[n]{x}$ where $n$ is an even integer is defined for $x\geq0$ [because $\sqrt[n]{x}$ for $x<0$ is imaginary].

5.$f(x)=\sqrt[m]{x}$ where $m$ is an odd integer is defined for every value of $x$.

6. $f(x)=\log_{a}x$ where $a>0$ is defined only for $x>0$. For negative values of $x$, this function does not exist [in fact, it becomes imaginary].

7. $f(x)=a^{x}$ where $a>0$; that is, the exponent is a variable and the number $a$ being a constant. This function is defined for every value of $x$.

8.$f(x)=\sin x$ or $f(x)=\cos x$ is defined for every value of $x$.

9. $f(x)=\tan x$ is defined for every value of $x$ except $x=(2n+1)\frac{\pi}{2}$ where $n$ is any integer.Recall that $\tan x=\frac{\sin x}{\cos x}$. Because $\cos\left((2n+1)\frac{\pi}{2}\right)=0$, and division by zero is not allowed, these values must be excluded here.

10. $f(x)=\arcsin x$ or $f(x)=\arccos x$ is defined only for $-1\leq x\leq1$.Because sines and cosines cannot exceed +1 or become less than –1, it follows that these functions are defined for all values of $x$ ranging from –1 to +1 inclusive, but not for other values.

11.$f(x)=\arctan x$ or $f(x)=\text{arccot }x$ is defined for every value of $x$.

When we talk about “the function $y=f(x)$” and the domain of the function is not specified, the domain is then assumed to be the set of all real numbers for which $f(x)$ is defined; that is, $f(x)$ is real. This set is sometimes called the natural domain of $f$ and is denoted by $Dom(f)$.

*all*real numbers for which $f(x)$ is real is called the

**natural domain**or simply the domain of the function.

- If we want to restrict the domain of a function, we must specify so. For example, we must write “$f(x)=x^{2},x>0$”, to restrict the domain of $f$ to positive numbers. If we do not add “$x>0$”, the domain of $f$ is understood to be the entire set of real numbers.

Another important set that is associated with every function is the range of a function. The range of a function $f$ is the set of all values taken by the function (or equivalently by the dependent variable) when the independent variable varies over the domain. In other words, all possible outputs of a function make up its range. The range of $f$ is denoted by $Rng(f)$.

\[

Rng(f)=\{f(x)|\ x\in A\}.

\]

- Note that the range of $f$ is a subset of its co-domain $B$

\[

Rng(f)\subset B.

\] - The difference between domain, codomain and the range of a function is depicted in Figure 3.