Case 1:
$$\frac{2x^2}{4x^3}$$
$$\frac{1}{2x}$$
The graph hasn't changed.
Case 2:
$$\frac{6t^4+4t^2}{t}$$
$$6t^3+4t$$
The graph hasn't changed.
Case 3:
$$\frac{x^3-8}{x^2-4}$$
$$\frac{(x-2)(x^2+2x+4)}{(x+2)(x-2)}$$
$$\frac{(x^2+2x+4)}{(x+2)}$$
The graph hasn't changed.
Question(s):
- How will I know that the graph will be the same after my operations?
- What things should I always avoid if I want to keep the function/graph the same?
For case $1$ we have that the two expressions are indeed equivalent since they are defined in both cases for $x\neq 0$.
For case $2$ the function are different because the first expression is not defined at $t=0$.
For case $3$ the function are different because the first expression is not defined at $x=2$.
More in general for $b(x_0)\neq 0$
$$f(x)=\frac{(x-x_0)a(x)}{(x-x_0)b(x)} \quad g(x)=\frac{a(x)}{b(x)}$$
have the same graph on all the domain of definition but $f(x)$ is not defined at $x=x_0$. The two expressions are equivalent if also $g(x)$ is not defined at $x=x_0$.
Refer also to the related