Q. 2 - Calculus

Question

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 $\left[a,b\right]$.

 (c) An increasing function on $\left[a,b\right]$.

Step-by-Step Solution

Answer

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

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

(c) If the function is increasing on $\left[a,b\right]$, if and only if for any $x_1<x_2$ and in the interval $\left[a,b\right]$. Then $f\left(x_1\right)<f\left(x_2\right)$.

Step 1. Given Information.

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

 A positive function on $\left[a,b\right]$.

 An increasing function on $\left[a,b\right]$.

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 $\left[a,b\right]$, if and only if $f\left(x\right)\ge 0$, for any $x$in the interval $\left[a,b\right]$.

Step (c). Definition of the function.

If the function is increasing on $\left[a,b\right]$, if and only if for any  $x_1<x_2$and in the interval $\left[a,b\right]$. Then $f\left(x_1\right)<f\left(x_2\right)$.