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 $Dom(f)$.
- 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.
(b)We recall that $\sqrt{a}$ is real when $a\geq0$. Thus $g(x)$ is defined when $x-1\geq0$ or $x\geq1$. That is, $Dom(g)=\{x|\ x\geq1\}=[1,\infty).$
(c)$h(x)$ is defined when $2-x\geq0$ or $x\leq2$. Thus $Dom(h)=\{x|\ x\leq2\}=(-\infty,2]$.
(d) Since $x^{2}-6x$ or $x(x-6)$ must not be negative, $x$ and $x-6$ must always have the same signs (see the following sign table). $F(x)$ is defined for every value of $x$ except those between 0 and 6; that is $Dom(F)=\{x|\ x\leq0,x\geq6\}.$ We can rewrite the domain of $F$ as $(-\infty,0]\cup[6,\infty)$.
(e) Since $4-x^{2}=(2-x)(2+x)$ must not be negative, $G(x)$ is defined where $2-x$ and $2+x$ have the same signs (see the following sign table). Therefore, $G(x)$ is defined for $-2\leq x\leq2$, or $Dom(G)=\{x|\ -2\leq x\leq2\}=[-2,2]$.
(f) For $\sqrt{x-1}$ to be real, we must have $x-1\geq0$ or $x\geq1$. For $\sqrt{2-x}$ to be real, we must have $2-x\geq0$ or $x\leq2$. Thus, $H(x)$ is real, if we have both $x\geq1$ and $x\leq2$; that is, the domain of $H$ is $Dom(H)=\{x|\ 1\leq x\leq2\}=[1,2]$.
(g) Because $u(x)=\dfrac{1}{x^{2}-3x}=\dfrac{1}{x(x-3)}$, $u(x)$ is defined for all values of $x$ except $x=0$ and $x=3$. When $x=0$ or $x=3$, the divisor becomes zero and division by zero is not allowed. Thus $Dom(u)=\{x|\ x\neq0,\ x\neq3\}$. The domain of $u$ can also be represented by $\mathbb{R}-\{0,3\}$ or $(-\infty,0)\cup(0,3)\cup(3,\infty)$.
(h) Because the logarithm of a nonpositive number is not real, we must have $x-2>0$ or $x>2$. Thus $Dom(v)=\{x|\ x>2\}=(2,\infty)$.
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.
