2.8 Equal Functions

$f$ is defined for all values of $x$; that is, its domain is the entire set of real numbers $Dom(f)=\mathbb{R}$. The domain of $g$ is also the entire set of real numbers because $x^2$ is always nonnegative. Therefore the first condition for equality of two functions is satisfied:
$$Dom(f)=Dom(g).$$ However, we know $\sqrt{x^2}=|x|$ (see Properties of the $n$th root in Section1.7). Therefore,
$$g(x)=\sqrt{x^2}=|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}.$$ Because $f(x)\ne g(x)$ when $x$ is negative, these two functions are not equal. Graphs of $f$ and $g$ are illustrated in Figure 2(a,b).

 

Recall that the only numbers that we can plug into a logarithm are positive numbers (plugging negative numbers will give imaginary results).Therefore, the function $f$ is defined for $x>0$; that is, $Dom(f)=\{x|x>0\}=(0,\mathrm{\infty })$.
Because always $x^2>0$ except when $x=0$, the domain of $g$ is $Dom(g)=\{x|x\ne 0\}=(\mathrm{\infty },0)\cup (0,\mathrm{\infty })$, which can also be rewritten as $\mathbb{R}\mathbb{-}\{0\}$. Note that $\ln (0)$ is not defined.
Because
$$Dom(f)\ne Dom(g),$$ these two functions are not equal. However, the two functions will become equal if we restrict the domain of $g$ to positive numbers.
That is,
$$2\ln x=\ln x^2\text{when }x>0.$$ Graphs of $f$and $g$ are depicted in Figure 3.