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What are the different ways of finding the domain of any function, with special emphasis on polynomial, rational, logarithmic, and exponential functions ?

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The rules that will get you through most introductory courses are

  1. You can't divide by zero,

  2. You can't take a square root of a negative number, and

  3. You can't take the logarithm of a non-positive number.

So, if someone hands you a function and asks you about its domain (which I am interpreting as meaning, find all real $x$ for which the function is defined), first see whether it has any quotients in it. If it does, find all the places where the denominator is zero --- those will have to be excluded from the domain. Note that if there's a tangent, cotangent, secant, or cosecant, then there's a quotient, e.g., $\tan x={\sin x\over\cos x}$.

Next, see whether there are any square roots (4th roots, 6th roots, 8th roots, etc., all count as square roots for these purposes, but not 3rd roots, 5th roots, 7th roots, etc.). If there are, then you have to exclude from the domain any values of $x$ which would result in the extraction of the square root of a negative number.

Finally, see whether there are any logarithms. If so, you'll have to exclude from the domain any values of $x$ which would result in the calculation of a logarithm of a nonpositive number.

Once you've made all the applicable exclusions, what's left of the real line is the domain.

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The natural maximal domain of definition for the following real-defined functions is:

(1) Polynomials: The whole real line

(2) Rational functions: The whole real line minus the roots of the denominator polynomial

3) Logarithmic function: the positive real numbers

4) Exponential functions: the whole real line.

If you had in mind functions that are not real-defined then tell us.

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Thanks. No, I was only asking about functions defined on real numbers.But the thing is how do you apply all these rules to composite function, maybe, involving multiplication, division and addition/subtraction of these functions. – wamiq reyaz Nov 29 '12 at 12:19
Sometimes in cases of composed functions, the necessary argument will have to be fairly tricky. Just make sure that for each given $x$, the successive steps all are well-defined. – Lubin Nov 30 '12 at 3:11

For a polynomial, the domain is all real numbers. For a rational function, you must solve for when the denominator is not equal to zero as a restriction. If you have a square root, you must set what's under the radical to be greater than or equal to zero. If you have a square root in the denominator of a rational function, you set what's under the root to be greater than zero (not equal to) because zero is undefined in the denominator. If you have a square root in the numerator of a rational function, you have two restrictions: one from setting what's under the square root to be greater than or equal to zero, and a second from solving for when the denominator is not equal to zero. Then finally in this case you'd have to take the intersection of your two restrictions.

For logs, the domain is only positive real numbers. For more help with how to find the domain, here's a video that explains it:

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