Finding the domain of a function...?

so the domain is basically every x value a function can have, so in this case if you factor x^2-4 into (x+2)(x-2), x cannot be + or - 2 because that means you are dividing by 0 in the denominator which is undefined. So in this case the domain is all real numbers except for 2 or -2

the domain is all the values of X that will result in a valid answer

start with the assumption that all values of x will work and then find the exceptions.

what's the problem with the function? how could you possibly put in any number and not get an answer?

like if I put in 1 you'd get:

2(1)

-----

1^2-4

which is -2/3, so no problem there

but see what happens if you put in 2:

2(2)

-----

2^2-4

which is 2/0

what is 2 divided by 0? ask your calculator? it'll give you error

you can't divide by 0, well technically they say it's undefined because no one has ever found a way to define it but for all practical purposes it's not possible to have 0 in the denominator of a fraction

but if you just answer 2 you'd be wrong because there's another number you can't use

try -2

2(-2)

-----

(-2)^2-4

when you square -2 you get 4 (positive) also

so neither -2 nor 2 are in the domain of the function

in this question you can just try those two obvious numbers but in the future you'll be given hard ones so trial and error isn't a good idea

instead, write an equation with the denominator = 0 and solve it

x^2-4=0

x^2=4

now take the square root of both sides

x = +/- 2

+/- meaning both plus and/or minus

your calculator will just say 2, so it's up to you to know that when you take the square root of a number that the negative is also a possible answer. Your calculator probably isn't smart enough to consider that.

so the domain is all numbers except -2 and 2, you can write it in a number of different ways but the most common is interval notation:

(-inf,-2)U(2,inf)

where 'inf' is the infinity symbol

Remember: the domain of a function is the set of real numbers of x for which the value of the function is real.

The value of h(x) = 2x/(x^2-4) is NOT real when the denominator is zero. So you've got to find the values of x that would make the denominator of h(x) equal to zero, then exclude these values of x from the domain.

Thus,

x^2 - 4 = 0

(x + 2)(x - 2) = 0

x + 2 = 0; x = -2

x - 2 = 0; x = 2

Answer:

The domain of h(x) are all real numbers, except x = -2 and x = 2.

OR

(-∞, -2) Ụ (-2, 2) Ụ (2, ∞)

h(x) = 2x/(x^2-4) = 2x/(x+2)(x-2)

Now, (x+2)(x-2) cannot equal zero since it would make the expression undefined by dividing by zero, so x cannot equal 2 or -2. Otherwise, all other values for x are OK when used in the function h(x).

Hence, the domain is all real numbers x such that x does not equal 2 or -2.

Your expression could be:

h(x) = 2*x/(x^2 - 4), factorizing the denominator x^2 - 4 = (x + 2)*(x - 2). It means that x must be different to + 2 and - 2, because these values do undetermined the function.

Domain: all the real numbers, ℝ, except x = + 2, x = -2