Question

graph the following function. what are the domain and range of the function? describe the intervals over which the function is positive and over which it is negative. p(x) = -5|x| use the graphing tool to graph the function. click to enlarge graph what is the domain of the function? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the domain is {x|x is a real number}. b. the domain is discrete. the domain is { } (type your answer(s) as integers or simplified fractions. use a comma to separate answers as needed)

Explanation:

Step1: Analyze the domain of \(p(x)=-5|x|\)

The absolute - value function \(y = |x|\) is defined for all real values of \(x\). When we multiply it by \(-5\) to get \(p(x)=-5|x|\), there are no restrictions on the input \(x\). So, for any real number \(x\), we can calculate \(p(x)\).

Step2: Analyze the range of \(p(x)=-5|x|\)

Since \(|x|\geq0\) for all \(x\in R\). Multiply both sides of the inequality \(|x|\geq0\) by \(- 5\). When we multiply an inequality by a negative number, the direction of the inequality sign changes. So, \(-5|x|\leq0\). When \(x = 0\), \(p(0)=-5|0| = 0\).

Step3: Analyze where the function is positive or negative

Set \(p(x)=-5|x|>0\). Then \(|x|<0\). But the absolute - value function \(y = |x|\) is non - negative (\(|x|\geq0\) for all \(x\in R\)), so there is no \(x\) such that \(p(x)>0\).
Set \(p(x)=-5|x|<0\). Then \(|x|>0\). The solution of \(|x|>0\) is \(x
eq0\).

Answer:

• Domain: A. The domain is \(\{x|x\text{ is a real number}\}\)
• Range: The range is \(\{y|y\leq0\}\)
• Positive interval: The function \(p(x)=-5|x|\) is never positive.
• Negative interval: The function \(p(x)=-5|x|\) is negative for the interval \((-\infty,0)\cup(0,\infty)\)