Chapter 4, Problem 1CRQ

(a)

To determine

To fill: “A function f is increasing on an interval I for any two numbers $x_1$ and $x_2$ in I, $x_1<x_2$ implies that ____.”

Answer to Problem 1CRQ

A function f is increasing on an interval I for any two numbers $x_1$ and $x_2$ in I, $x_1<x_2$ implies that $f\left(x_1\right)<f\left(x_2\right)$.

Explanation of Solution

A function f is said to be increasing on an interval $\left(a,b\right)$ if for every two numbers $x_1$ and $x_2$ in the interval $\left(a,b\right)$ is such that $f\left(x_1\right)<f\left(x_2\right)$, whenever $x_1<x_2$.

The graph of a function f is shown below in Figure 1.

Figure 1

The function in Figure 1 shows two numbers $x_1$ and $x_2$ in the interval $\left(a,b\right)$ where $x_1<x_2$ along with their corresponding function values $f\left(x_1\right)<f\left(x_2\right)$.

(b)

To determine

To fill: “A function f is decreasing on an interval I for any two numbers $x_1$ and $x_2$ in I, $x_1<x_2$ implies that ____.”

Answer to Problem 1CRQ

A function f is decreasing on an interval I for any two numbers $x_1$ and $x_2$ in I, $x_1<x_2$ implies that $f\left(x_1\right)>f\left(x_2\right)$.

Explanation of Solution

A function f is said to be decreasing on an interval $\left(a,b\right)$ if for every two numbers $x_1$ and $x_2$ in the interval $\left(a,b\right)$ is such that $f\left(x_1\right)>f\left(x_2\right)$ whenever $x_1<x_2$.

The graph of a function f is shown below in Figure 2.

Figure 2

The function in Figure 2 shows two numbers $x_1$ and $x_2$ in the interval $\left(a,b\right)$ where $x_1<x_2$ along with their corresponding function values $f\left(x_1\right)>f\left(x_2\right)$.