Chapter 3.5, Problem 4CFU

To determine

To write: a paragraph on comparing the monotonicity of the given functions, and make a conjecture about the monotonicity of the reflection over the x-axis of any function as compared that of the original function.

Answer to Problem 4CFU

Thus, on reflecting the monotonic $f\left(x\right)$ over the x-axis, another monotonic function $g\left(x\right)$ with change in monotonic direction will be formed.

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=x^2\\ g\left(x\right)=-x^2\end{array}$

Calculation:

The graph of $f\left(x\right)$ and $g\left(x\right)$ represented by red and blue color respectively is shown below.

  

The graph of $f\left(x\right)$ and $g\left(x\right)$ are not monotonic because they both increase and decrease in their domain. However, $f\left(x\right)$ is monotonic from $-\infty \text{ to 0}$ as it is constantly decreasing. Also $g\left(x\right)$ is monotonic from $-\infty \text{ to 0}$ as it is constantly increasing. Thus, on reflecting the monotonic $f\left(x\right)$ over the x-axis, another monotonic function $g\left(x\right)$ with change in monotonic direction will be formed.