Chapter 1.2, Problem 83E

Solution

(a)

To find: the least upper bound and tell whether or not it is in the range of the function.

The least upper bound is $2$ .

Given information:

The function $f\left(x\right)=2-0.8x^2$ .

Calculation:

Here,

  $0.8x^2\ge 0$ , therefore $2-0.8x^2\le 2$ .

The least upper bound is in the range because $f\left(0\right)=2$ .

(b)

To find: the least upper bound and tell whether or not it is in the range of the function.

Least upper bound: 3

Not in the range of $g$ .

Given information:

The function $g\left(x\right)=\frac{3x^2}{3+x^2}$ .

Calculation:

The graph is $g\left(x\right)=\frac{3x^2}{3+x^2}$

  

The least upper bound of the function is $3$ . It is a horizontal asymptote; it is not in the range of $g$

(c)

To find: the least upper bound and tell whether or not it is in the range of the function.

No upper bound.

Given information:

The function $h\left(x\right)=\frac{1-x}{x^2}$ .

Calculation:

The graph of $h\left(x\right)=\frac{1-x}{x^2}$

  

The function approaches infinity as $x$ approaches zero, which means it has

no upper bound.

Therefore, no upper bound.

(d)

To find: the least upper bound and tell whether or not it is in the range of the function.

Least upper bound: 2

In the range of $p$ .

Given information:

The function $p\left(x\right)=2\sin \left(x\right)$ .

Calculation:

The graph of $p\left(x\right)=2\sin \left(x\right)$

  

The least upper bound is $2$ . It is part of the range of $p$ , since $p\left(x\right)=2$ when $x=\pi /2+2\pi k$ for all integers $k$ .

Hence, least upper bound: 2

In the range of $p$ .

(e)

To find: the least upper bound and tell whether or not it is in the range of the function.

Least upper bound: $1$

In the range of $q$ .

Given information:

The function $q\left(x\right)=\frac{4x}{x^2+2x+1}$ .

Calculation:

The graph of $q\left(x\right)=\frac{4x}{x^2+2x+1}$

  

The least upper bound as $1$ . It is part of the range of $q$ , since

  $\begin{array}{c}q\left(1\right)=\frac{4}{1+2+1}\\ =1\end{array}$

Hence, least upper bound: $1$

In the range of $q$ .