You can put this solution on YOUR website! Sometimes you have the domain of a function given to you. Sometimes you have to figure it out for yourself. When you find the domain yourself you
Start by assuming that the domain is all real numbers; then
Rule out any x values that would create an expression "that should not be". And what are these expressions?:
Division by zero. This includes obvious divisions by zero like a denominator that is zero. And less obvious ones: tan or sec of odd multiples of radians (or 90 degrees) and cot or csc of even multiples of radians (or 90 degrees).
Negative radicands of even-numbered roots. "Radicand" is the name for the expression inside a radical.) Even-numbered roots are square roots, 4th roots, 6th roots, etc. Radicands of these roots cannot be negative because you cannot raise a real number to an even power and get a negative result.
Zero or negative arguments of logarithms.
Bases of logarithms that are zero, one or any negative number.
Any other expression that you learn :should never be".
If your function has none of these types of expressions, then your domain is all real numbers.
Your function does have a denominator (but none of the other types of expressions that can limit a domain). So all we have to do is make sure the denominator is never zero. To do this we set the denominator to zero, solve for x and then we know what values x cannot have:
This is a quadratic equation. TO solve it, we want one side to be zero (which we already have) and then we factor the other side (or use the Quadratic Formula). This expression factors fairly easily:
From the Zero Product Property we know that one of these factors must be zero. So:
x-4 = 0 or x+2 = 0
Solving these we get:
x = 4 or x = -2
These are the values x cannot be! So the domain is all real numbers except 4 or -2.
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