- Set the denominator \(x^2 - 9\) equal to zero to find where the expression is undefined.
- Factor the expression as \((x-3)(x+3)=0\) to solve for \(x\).
- Identify that the expression is undefined at \(x=3\) and \(x=-3\).
- The only correct option from the provided list is \(\boxed{x
eq 3}\).
Explanation
- Understanding the Domain of a Rational Expression
A rational expression is a fraction where both the numerator and the denominator are polynomials. A fundamental rule in mathematics is that division by zero is undefined. Therefore, a rational expression is undefined at any value of \(x\) that makes its denominator equal to zero. To find the domain restrictions, we must set the denominator of the expression \(\frac{x^2+5x+6}{x^2-9}\) equal to zero and solve for \(x\).
- Identifying the Restrictions
The denominator of the given expression is \(D(x) = x^2 - 9\). We set this equal to zero to identify the values for which the expression is undefined:
\[x^2 - 9 = 0\]
This equation can be solved by factoring the difference of squares:
\[(x - 3)(x + 3) = 0\]
Applying the zero product property, we have two possibilities:
\[x - 3 = 0 \implies x = 3\]
\[x + 3 = 0 \implies x = -3\]
Thus, the expression is undefined at \(x = 3\) and \(x = -3\). These are the domain restrictions.
- Selecting the Correct Restriction
Comparing our results to the given options:
- \(x
eq 3\) (This matches one of our found restrictions)
- \(x
eq 2\) (Not a restriction)
- \(x
eq 1\) (Not a restriction)
- \(x
eq -1\) (Not a restriction)
- \(x
eq -2\) (Not a restriction)
While we also identified \(x
eq -3\) as a restriction, it is not listed among the given options. Therefore, the only applicable restriction provided in the list is \(x
eq 3\). Note that even if a factor cancels out (like \((x+3)\) does in the numerator and denominator), the original expression remains undefined at that point.
Examples
Domain restrictions are critical in fields like engineering and economics, where formulas often model real-world behavior only within specific bounds. For instance, if you are designing a bridge, the load capacity formula might have a denominator that becomes zero at a specific weight, signaling a point of structural failure. Similarly, in economics, a cost function might be undefined for a negative production level, which is physically impossible. Identifying these "undefined" points helps prevent errors and ensures models remain valid within their intended scope.