Question

problem 7
here are three statements:

• the function is always positive.
• the function is always increasing.
• the function is always decreasing.

sketch a graph of a function so that two of the
statements are true and one is false.

Explanation:

Step1: Analyze the "always positive" statement

A function is always positive if \(y>0\) for all \(x\) in its domain. For example, \(y = 2\) is a constant function (a special case of non - decreasing and non - increasing). But we can also consider a function like \(y=e^{x}\), which is always positive.

Step2: Analyze the "always increasing" statement

A function \(y = f(x)\) is always increasing if for any \(x_10\), so it is always increasing.

Step3: Analyze the "always decreasing" statement

A function \(y = f(x)\) is always decreasing if for any \(x_10\) and \(y = e^{x}>0\) for all \(x\in R\)), it cannot be always decreasing.

We can sketch a function like \(y = e^{x}\). The domain of \(y = e^{x}\) is \((-\infty,\infty)\), the range is \((0,\infty)\) (so it is always positive), and its derivative \(y^\prime=e^{x}>0\) (so it is always increasing). It is not always decreasing.

Answer:

Sketch a graph of \(y = e^{x}\). The function \(y = e^{x}\) is always positive (since \(e^{x}>0\) for all \(x\in R\)) and always increasing (since its derivative \(y^\prime = e^{x}>0\) for all \(x\in R\)), and it is not always decreasing.