Question

1. an angle critical thinking 49. when a fraction contains a variable in the denominator, there are some values of the variable for which the fraction is undefined. find the domain of ( f(x) = \frac{15}{x^2 - 9} ). what value(s) of x would ( f(x) = 0 )? on a graph of

Explanation:

Step1: Identify when the fraction is undefined

A fraction is undefined when its denominator is zero. So, we set the denominator equal to zero and solve for \(x\):
\(x^2 - 9 = 0\)

Step2: Solve the equation for \(x\)

We can factor the left - hand side using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\), where \(a=x\) and \(b = 3\). So, \(x^2-9=(x + 3)(x - 3)=0\)
If \((x + 3)(x - 3)=0\), then by the zero - product property, either \(x+3 = 0\) or \(x - 3=0\)

• For \(x+3 = 0\), we get \(x=-3\)
• For \(x - 3=0\), we get \(x = 3\)

The domain of a function is the set of all real numbers except the values that make the function undefined. So, the domain of \(f(x)=\frac{15}{x^2 - 9}\) is all real numbers except \(x=-3\) and \(x = 3\). In interval notation, the domain is \((-\infty,-3)\cup(-3,3)\cup(3,\infty)\)

Answer:

The domain of \(f(x)=\frac{15}{x^2 - 9}\) is all real numbers except \(x=-3\) and \(x = 3\), or in interval notation \((-\infty,-3)\cup(-3,3)\cup(3,\infty)\)