Question

find the domain of the function $g(x) = \frac{6}{4x - x^2}$. write both set - builder notation and interval notation for the answer.

write the domain in set - builder notation. select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. the domain of $g$ is $\\{x|x$ is a real number and $x\
eq \square\\}$.
(type an integer or a fraction. use a comma to separate answers as needed.)

b. the domain of $g$ is $\\{x|x$ is a real number and $x = \square\\}$.
(type an integer or a fraction. use a comma to separate answers as needed.)

c. the domain of $g$ is $\\{x|x$ is a real number\\}.

the domain of $g$ is $\square$
(type your answer in interval notation.)

Explanation:

Step1: Find when denominator is zero

To find the domain, we need to determine when the denominator \(4x - x^2\) is zero (since division by zero is undefined). So, solve the equation \(4x - x^2 = 0\). Factor out \(x\): \(x(4 - x)=0\).

Step2: Solve for x

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So, \(x=0\) or \(4 - x = 0\). Solving \(4 - x = 0\) gives \(x = 4\).

Step3: Determine set - builder notation

The domain of a function is all real numbers except the values that make the denominator zero. So, the domain of \(g(x)\) in set - builder notation is \(\{x|x\text{ is a real number and }x
eq0,4\}\). So, option A is correct, and we fill in \(0,4\) in the box.

Step4: Determine interval notation

The domain in interval notation is \((-\infty,0)\cup(0,4)\cup(4,\infty)\).

Answer:

For set - builder notation: A. The domain of \(g\) is \(\{x|x\text{ is a real number and }x
eq\boldsymbol{0,4}\}\)
For interval notation: The domain of \(g\) is \(\boldsymbol{(-\infty,0)\cup(0,4)\cup(4,\infty)}\)