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Section 3.4 : The Definition of a Function

15. Determine the domain of the following function.

\[f\left( x \right) = \frac{1}{{{x^2} - 25}}\] Show Solution

Recall that the domain is simply all the values of \(x\) that we can plug into the function and the resulting function value will be a real number.

In this case we have a rational expression where the numerator is a constant (and so won’t cause any problems for any value of \(x\)) and the denominator is a polynomial.

So, we need to avoid division by zero for this problem. We will get division by zero at,

\[{x^2} - 25 = 0\hspace{0.25in} \to \,\,\ {x^2} = 25\hspace{0.25in} \to \,\,\ x = \pm 5\]

Therefore, the domain for this function is all real numbers except \(x = \pm 5\) .