In precalculus, you might encounter expressions like $\frac{x^2 + x - 6}{x + 3} = x - 2$. While these two expressions appear similar, they do not define the same function for all values of $x$.

Why They Are Not the Same Function

1. Domain Differences:

   • The function $f(x) = \frac{x^2 + x - 6}{x + 3}$ has a restriction on the domain because $x + 3$ appears in the denominator. When $x = -3$, the denominator becomes zero, which makes $f(x)$ undefined at $x = -3$.
   • In contrast, $g(x) = x - 2$ has no such restriction, as there is no denominator that could cause a division by zero. Thus, $g(x)$ is defined for all real numbers.

2. Simplification Process:

   • The expression $\frac{x^2 + x - 6}{x + 3}$ can be simplified to $x - 2$ for values of $x \neq -3$, because $x^2 + x - 6$ factors to $(x + 3)(x - 2)$, allowing us to cancel $x + 3$ from the numerator and denominator