Question

find the the domain of the function $f(x) = \frac{x + 5}{x^2 + 11x - 12}$.
\bigcirc \\{x| x \
eq 5\\}
\bigcirc \\{x| x \
eq 12 \text{ and } x \
eq -5\\}
\bigcirc \\{x | x \
eq -12 \text{ and } x \
eq 1\\}
\bigcirc \\{x| x \
eq -12\\}
\bigcirc \\{x| x \
eq 12\\}
express the solution interval notation \boxed{}
question help: \boxed{\text{video}} \boxed{\text{written example}}

Explanation:

Step1: Factor the denominator

To find the domain of a rational function, we need to determine the values of \( x \) that make the denominator zero (since division by zero is undefined). So, we factor the quadratic in the denominator: \( x^2 + 11x - 12 \). We need two numbers that multiply to \( -12 \) and add to \( 11 \). Those numbers are \( 12 \) and \( -1 \). So, \( x^2 + 11x - 12=(x + 12)(x - 1) \).

Step2: Find the values that make denominator zero

Set the factored denominator equal to zero: \( (x + 12)(x - 1)=0 \). Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So, \( x+12 = 0 \) gives \( x=-12 \) and \( x - 1=0 \) gives \( x = 1 \).

Step3: Determine the domain

The domain of the function \( f(x)=\frac{x + 5}{x^2+11x - 12} \) is all real numbers except the values that make the denominator zero. So, the domain is \( \{x|x
eq - 12\text{ and }x
eq1\} \).

Step4: Express in interval notation

The interval notation for the domain is \( (-\infty,-12)\cup(-12,1)\cup(1,\infty) \).

Answer:

The correct option for the domain in set - builder notation is \( \{x\mid x
eq - 12\text{ and }x
eq1\} \). The interval notation is \( (-\infty,-12)\cup(-12,1)\cup(1,\infty) \).