Why simplifying this function generates different outputs for same input?

[gen x]

This function $f(x) = \frac{x^2 - 4}{x - 2}$ when simplify become $f(x) = x + 2$ .

If I enter $x=2$ in the first expression I get $y=\frac{0}{0}$, and if I enter $x=2$ in the second expression I get $y=4$.


Why different results if this is same function, just simplifly?

 

[kuruman]

Why different results if this is same function, just simplifly?

Because you cannot "just simplify" unless you specifically exclude $x=2$ at which point, as you have discovered, the function is undefined. In other words, the two functions are "the same" except at $x=2.$

What would you say to someone who argues that, for any $a$, $$f(2)=\frac{2^2-4}{2-2}=\frac{0}{0}=\frac{0\cdot a}{0}=\frac{\cancel{0}\cdot a}{\cancel{0}}=a~~?$$

 

[Mark44]

This function $f(x) = \frac{x^2 - 4}{x - 2}$ when simplify become $f(x) = x + 2$ .

If I enter $x=2$ in the first expression I get $y=\frac{0}{0}$, and if I enter $x=2$ in the second expression I get $y=4$.


Why different results if this is same function, just simplifly?

The two functions are identical except at the point (2, 4). The graph of $f(x) = \frac{x^2 - 4}{x - 2}$ can be thought of as a straight line whose slope is 1 and with a y-intercept of 2, but with a "hole" at (2, 4).

 

[FactChecker]

Good catch! You did very well to check the final answers in the original formula. As you go through a simplification, you should keep track of any values of x where the original function was undefined. That can become difficult and tricky. It is wise to check your results in the original formula.

 

[Mark44]

As you go through a simplification, you should keep track of any values of x where the original function was undefined.

An important attribute of any function is its domain. The difference between the two functions in the OP is that, because of the division, the domain can't include any number that causes division by zero. Similar restrictions occur with some of the trig functions, log functions, and so on.

 

[gen x]

OK thanks for answer, I always thought that simplify function doesn't change absolute nothing..

 

[willyengland]

I agree and remember that I found this mysterious back then.
Think of it as "dividing by zero", which is forbidden. So, this simplification is forbidden.

 

[Dale]

So, this simplification is forbidden.

It is forbidden specifically at $x=2$. Elsewhere it is not forbidden.

 

[FactChecker]

OK thanks for answer, I always thought that simplify function doesn't change absolute nothing..

Correct simplification includes keeping track of where the original function was undefined.