Question

find the domain of the function.
f(x)=\frac{1}{\frac{16}{x + 3}-2}
the domain of f(x) is .
(type your answer in interval notation.)

Explanation:

Step1: Identify values making denominator zero

The denominator of the function is $\frac{16}{x + 3}-2$. Set it equal to zero: $\frac{16}{x + 3}-2=0$.

Step2: Solve the equation for x

First, add 2 to both sides: $\frac{16}{x + 3}=2$. Then cross - multiply: $16 = 2(x + 3)$. Expand the right side: $16=2x+6$. Subtract 6 from both sides: $10 = 2x$. Divide by 2: $x = 5$. Also, the expression $\frac{16}{x + 3}$ is undefined when $x+3=0$, i.e., $x=-3$.

Step3: Determine the domain

The domain is all real numbers except $x=-3$ and $x = 5$. In interval notation, it is $(-\infty,-3)\cup(-3,5)\cup(5,\infty)$.

Answer:

$(-\infty,-3)\cup(-3,5)\cup(5,\infty)$