document.write( "Question 202423: State the domain of the following and provide a brief explanation of the answer.\r
\n" ); document.write( "\n" ); document.write( "m(x)=5/x^2-9\r
\n" ); document.write( "\n" ); document.write( "l(x)=5x-4\r
\n" ); document.write( "\n" ); document.write( "g(x)=7x=4/x=4 \n" ); document.write( "
Algebra.Com's Answer #152698 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
The domain of a function is the set of all possible values for the input variable, which is usually \"x\". Generally, when a domain is not explcitly defined, the domain of a function is all Real numbers. However, one must exclude values which cause expressions which cannot be allowed.

\n" ); document.write( "Examples of expressions which cannot be allowed:
  • Division by zero
  • Negative radicands of even-numbered roots:
    • \"g%28x%29+=+sqrt%28x+%2B+3%29\" Since there are no square roots of negative numbers (within the set of Real numbers) the domain of g must ensure that (x+3) >= 0. In other \"words\", x >= -3.
    • \"h%28x%29+=+root%286%2C+3x+-+6%29\". Since there are no 6th roots of negative numbers (within the set of Real numbers) the domain of h must ensure that (3x - 6) >= 0. In other \"words\", x >= 2.
    • Note that \"q%28x%29+=+root%283%2C+4x+%2B+9%29\" has a domain of all real numbers since cube roots (in fact all odd-numbered roots) of negative numbers do exist within the set of Real numbers.
  • Negative or zero agruments to logarithm functions (regardless of the base of the logarithm).
  • Arguments which are not allowed by certain other functions which are part of the definition of the function in question. An example of this would be f(x) = tan(x) + 4. Since the tan function is not defined for 90 degrees (or \"pi%2F2\" radians), these values must be excluded from the domain of f(x).

\n" ); document.write( "Now let's apply this to your problems.

\n" ); document.write( "\"m%28x%29=5%2F%28x%5E2-9%29\"
\n" ); document.write( "Since we have a denominator we must avoid x-values that would make the denominator zero. So if we solve \"x%5E2+-+9+=+0\" we will find the x-values we must exclude from the domain. Factoring this equation we get \"%28x+%2B+3%29%28x+-+3%29+=+0\". From this we can see that the solution is x = 3 or x = -3.
\n" ); document.write( "So the domain of m(x) is all real numbers except 3 and -3.
\r
\n" ); document.write( "\n" ); document.write( "l(x)=5x-4
\n" ); document.write( "Since none of the items described above (denominators, even-numbered roots, logarithms, etc.) are present, there is nothing to exclude. The domain is all Real numbers.
\r
\n" ); document.write( "\n" ); document.write( "g(x)=7x=4/x=4
\n" ); document.write( "With 3 equal signs I'm not sure what this is. Since the \"=\" and the \"+\" are on the same key, I'm going to assume that the last two \"=\" are supposed to be \"+\".
\n" ); document.write( "If \"g%28x%29+=+7x+%2B+4%2Fx+%2B+4\" we must make sure the denominator of x does not become zero. So we must exclude 0 from the domain.
\n" ); document.write( "If instead \"g%28x%29+=+7x+%2B+4%2F%28x%2B4%29\" then we must make sure (x + 4) is not zero. So x must not be -4.
\n" ); document.write( " \n" ); document.write( "
\n" );