Q 0 - Calculus

Question

Functions: Provide definitions for each of the following:

•  function
•  the domain of a function
•  the codomain of a function
•  the function $f$ is increasing on interval $\left[a,b\right]$
•  the function $f$ is strictly increasing on interval $\left[a,b\right]$
•  the function $f$ is decreasing on interval $\left[a,b\right]$
•  the function $f$ is strictly decreasing on interval $\left[a,b\right]$
•  the function $f$ is constant on interval $\left[a,b\right]$
•  the function $f$ is bounded on interval $\left[a,b\right]$

Step-by-Step Solution

Answer

Definition of a function :-

The function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

Domain of a function :-

The domain of a function is defined as the set of all possible inputs for the function.

Codomain of a function :-

The codomain of a function is defined as the set of all possible outputs for the function.

Function $f$ is increasing on interval $\left[a,b\right]$ :-

A function $f$ is said to be increasing function when $f\left(x\right)\ge f\left(y\right) for x\ge y, where x,y\in \left[a,b\right]$.

Function $f$ is strictly increasing on interval  $\left[a,b\right]$ :-

A function $f$ is said to be strictly increasing function when $f\left(x\right)>f\left(y\right) for x>y, where x,y\in \left[a,b\right]$.

Function $f$ is decreasing on interval $\left[a,b\right]$ :- 

A function $f$ is said to be decreasing function when $f\left(x\right)\le f\left(y\right) for x\ge y, where x,y\in \left[a,b\right]$.

Function $f$ is strictly decreasing on interval $\left[a,b\right]$ :-

 A function $f$ is said to be strictly decreasing function when $f\left(x\right)<f\left(y\right) for x>y, where x,y\in \left[a,b\right]$.

Constant Function :-

A function $f$ is said to be a constant function over interval $\left[a,b\right]$ if the value of the function remains same for every value on interval $\left[a,b\right]$. 

Bounded Function :-

A function is said to be a bounded function if there exists numbers $a and A$such that $a\le f\left(x\right)\le A, for all x\in \left[a,b\right]$.

Step 1. Given Information

We have to write the definition of a function, the domain of a function, the codomain of a function, the function $f$ is increasing on interval $\mathrm{[}\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{]}$, the function $f$ is strictly increasing on interval $\left[a,b\right]$, the function $f$ is decreasing on interval $\left[a,b\right]$, the function $f$ is strictly decreasing on interval $\left[a,b\right]$, the function $f$ is constant on interval $\left[a,b\right]$ and the function $f$ is bounded on interval $\left[a,b\right]$.

Step 2. Definition of function

We know that a function is a special type of mathematical relation.

In this relation the every input value has only unique output. Also every input has output.

For example if we choose a set $X$ containing elements of type $x_i$ . Then we posses a relation $f$ on it. Then output of each element of $X$ is unique. That is each $f\left(x_i\right)$ is unique. Also there exist output for each element of $X$. Then the relation $f$ becomes a function.

From this information we can define a function as following :-

Function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

Step 3. Definition of a domain and codomain of a function

We know that a function is a defined from one set to other set. The elements of first set are called inputs and the elements of other set are called outputs.

The domain of a function is defined as the set of all possible inputs for the function.

Also The codomain of a function is defined as the set of all possible outputs for the function.

Step 4. Increasing and strictly increasing function over a interval

We know that increasing means the value is increased step by step.

So we can say that a function $f$ is an increasing function on interval $\left[a,b\right]$ if the value of a function $f$ is increasing when the input value increased over interval $\left[a,b\right]$.

That is $f\left(x\right)\ge f\left(y\right) for x\ge y, where x,y\in \left[a,b\right]$

The strictality holds when there is no equality.

So that a function $f$ is said to be strictly increasing function when $f\left(x\right)>f\left(y\right) for x>y, where x,y\in \left[a,b\right]$.

Step 5. Decreasing and strictly decreasing function over a interval

We know that decreasing means the value is decreased step by step.

So we can say that a function $f$ is a decreasing function on interval $\left[a,b\right]$ if the value of a function  is decreasing when the input value increased over interval .

That is $f\left(x\right)\le f\left(y\right) for x\ge y, where x,y\in \left[a,b\right]$

The strictality holds when there is no equality.

So that a function $f$ is said to be strictly decreasing function when $f\left(x\right)<f\left(y\right) for x>y, where x,y\in \left[a,b\right]$.

Step 6. Constant and bounded function

A function is constant means that the value are not varies.

So that a function $f$ is said to be a constant function over interval $\left[a,b\right]$ if the value of the function remains same for every value on interval $\left[a,b\right]$.

A function is bounded means that output values have boundaries that means the length of outputs is closed by the boundaries.

So that a function $f$ is said to be a bounded function if there exists numbers $a and A$, such that $a\le f\left(x\right)\le A, for all x\in \left[a,b\right]$.