# Find the domain of the function. Write your answer in both set builder and interval notation. f(x)= 8x / 2x^2 - x

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Find teh domain of `f(x)=(8x)/(2x^2-x)` :

The domain is the set of all possible inputs. Domain restrictions are typically division by zero, taking even roots of negative numbers, and taking logarithms of nonpositive numbers.

For this problem we are only concerned with division by zero.

The denonminator factors as x(2x-1). By the zero product property the denominator will be zero if x=0 or 2x-1=0 ==> `x=1/2` . Thus these values cannot be in the domain.

As this is a rational function, all other values of x are permissable.

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The domain is `{x|x in RR,x!=0,1/2}` or `x in (-oo,0)uu(0,1/2)uu(1/2,oo)`

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Domain = all valid x values.

`f(x) = (8x)/(2x^2-x)`

In functions with x in the denominator, one major idea to remember is that the value of x can never let the denominator equal zero.

First let's write this function in a more factored form.

`(8x)/(x(2x-1))`

We know that if x = 0 or if x = 1/2, the denominator will be zero. Therefore x cannot be these values.

Simplifying the function our result is:

`8/(2x-1)`

This is a transformation of the function `1/x`

So we know that the rest of the x values are valid (since the only invalid x value in 1/x is 0).

Therefore: The domain of x is all real numbers except for 0 and 1/2.

Set builder notation:

`{x|x!=0, x!=1/2}`

Interval notation:

`(-oo, 0) (0, 1/2) (1/2, oo)`

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