Question

select the correct answer.
which statement is true about the function shown in the graph?
a. the function is strictly increasing
b. the function is strictly decreasing
c. the function is increasing and decreasing
d. the function is constant

Explanation:

Step1: Analyze the function \(y = 2\sqrt{x + 4}\)

The domain of the function \(y = 2\sqrt{x+4}\) is \(x\geq - 4\) (since the expression under the square - root \(x + 4\geq0\)).
The derivative of the function \(y=f(x)=2\sqrt{x + 4}=2(x + 4)^{\frac{1}{2}}\) using the power rule \((x^n)^\prime=nx^{n - 1}\) is \(y^\prime=f^\prime(x)=\frac{2}{2\sqrt{x + 4}}=\frac{1}{\sqrt{x + 4}}\).

Step2: Determine the sign of the derivative

For \(x>-4\), \(\sqrt{x + 4}>0\), so \(y^\prime=\frac{1}{\sqrt{x + 4}}>0\).
When the derivative \(y^\prime=f^\prime(x)>0\) for all \(x\) in the domain \((x>-4)\) of the function \(y = 2\sqrt{x + 4}\), the function is strictly increasing.

Answer:

A. The function is strictly increasing