Chapter 4: Q. 2 (page 398)

Give formal mathematical definitions for each of the following functions.
(a) A function $f$ from a set $A$ to a set $B$ .
(b) A positive function on $[a, b]$ .
(c) An increasing function on $[a, b]$ .

Short Answer

Expert verified

(a) Every element of set $A$ is related to some other element of set $B$ .

(b) If the function is positive on $[a, b]$ , if and only if $f (x) \geq 0$ , for any $x$ in the interval $[a, b]$ .

(c) If the function is increasing on $[a, b]$ , if and only if for any $x_{1} < x_{2}$ and in the interval $[a, b]$ . Then $f (x_{1}) < f (x_{2})$ .

Step by step solution

Step 1. Given Information.

A function $f$ from a set $A$ to a set $B$ .

A positive function on $[a, b]$ .

An increasing function on $[a, b]$ .

Step (a). Definition of the function.

Every element of set $A$ is related to some other element of set $B$ .

No element in set $A$ is related to more than one element in set $B$ .

Step (b). Definition of the function.

If the function is positive on $[a, b]$ , if and only if $f (x) \geq 0$ , for any $x$ in the interval $[a, b]$ .

Step (c). Definition of the function.

If the function is increasing on $[a, b]$ , if and only if for any $x_{1} < x_{2}$ and in the interval $[a, b]$ . Then $f (x_{1}) < f (x_{2})$ .

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Give formal mathematical definitions for each of the following functions.
(a) A function $f$ from a set $A$ to a set $B$ .
(b) A positive function on $[a, b]$ .
(c) An increasing function on $[a, b]$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step (a). Definition of the function.

Step (b). Definition of the function.

Step (c). Definition of the function.

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Give formal mathematical definitions for each of the following functions.(a) A function ffrom a set Ato a set B.(b) A positive function on a,b.(c) An increasing function ona,b.

Short Answer

Step by step solution

Step 1. Given Information.

Step (a). Definition of the function.

Step (b). Definition of the function.

Step (c). Definition of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Applied Mathematics

Pure Maths

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Give formal mathematical definitions for each of the following functions.
(a) A function $f$ from a set $A$ to a set $B$ .
(b) A positive function on $[a, b]$ .
(c) An increasing function on $[a, b]$ .