Question

f(x) = 5 / (x - 2)
answer: x ≠ 2
My lesson does not tell me how they got that answer. I can not figure it out.

Answer 1

What do you think is the value of f(x) when x=2, or in short, what is f(2)?

Answer 2

if x = 2 then it would be f(x)= 5/0 which is just f(x) = 5?

Answer 3

Nope,
division by 0 is not defined
so
\[\frac{5}{0}\]
has no meaning, that's why x can take any value except 2
This is also the reason why you can't multiply both sides of an equation by 0
consider
\[y=2x\]
If you multiply both sides by 0 you get
\[y \times 0=2x \times 0\]
Now one may say to get back the original equation we simply have to divide by 0
\[\frac{y \times 0}{0}=\frac{2x \times 0}{0}\]
Now division by 0 is not defined, so this has no meaning, so you just simply can't multiply both sides of an equation by 0

Answer 4

Also remember
\[\frac{5}{1}=5\]
and you are saying
\[\frac{5}{0}=5\]
This implies that
\[1=0\]
which is not true

Answer 5

okay so since division by 0 is not defined, it would give x an infinite amount of values other than 2 because of how we worked out the equation? like x - 2 = 0 turns into x = 2 and when x is replaced by 2 we get 0 which brings us back to it being undefined?

Answer 6

Yes and this is what we call the "domain" of the function, the set of all the values that x can take
symbolically we write

Set of real numbers minus the set containing number 2

Answer 7

Basically x can take any value except 2 because for 2 it is not defined

Answer 8

thank you!!!