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 localid="1654357677089" $\left[a,b\right]$
• the function localid="1654357683495" $f$is strictly increasing on interval localid="1654357689590" $\left[a,b\right]$
• the function localid="1654357698687" $f$is decreasing on interval localid="1654357702734" $\left[a,b\right]$
• the function localid="1654357731613" $f$is strictly decreasing on interval localid="1654357708010" $\left[a,b\right]$
• the function localid="1654357726536" $f$is constant on interval localid="1654357712112" $\left[a,b\right]$
• the function localid="1654357721905" $f$is bounded on intervallocalid="1654357717212" $\left[a,b\right]$

Short 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 $f\left(x\right)\ge f\left(y\right)forx\ge y,wherex,y\in \left[a,b\right]$.

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

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

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

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

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

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

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$aandA$such that$a\le f\left(x\right)\le A,forallx\in \left[a,b\right]$.

Step-by-step solution

Step 1. Given Information

We have to write the definition of afunction,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 isa 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$f\left(x\right)\ge f\left(y\right)forx\ge y,wherex,y\in \left[a,b\right]$

The strictality holds when there is no equality.

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

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$f\left(x\right)\le f\left(y\right)forx\ge y,wherex,y\in \left[a,b\right]$

The strictality holds when there is no equality.

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

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 $aandA$, such that $a\le f\left(x\right)\le A,forallx\in \left[a,b\right]$.