What are functions?
Most simply, the functions in calculus are the way to describe the real world in mathematical terms like equations, tables or graphs.
Let see Here’s a plant, and what you see here is its shadow.
Can you list the things that the length of the shadow is dependent on?
Anything else you can think of?
- One, it is dependent on the position of the source of light.
- If the height of the plant grows, then the shadow’s length will also change, right?
So the length of the shadow is dependent on the position of the source of light and the height of the plant too.
So we can say that the length of the shadow is a FUNCTION of the following two things.
- Position of light
- Height of plant
The output is dependent on these two things which could be considered as the inputs.
Mathematical representation of function:
We know that when an object is in motion, it means its position changes with respect to time.
$$\text{Lets say} \;\;\; d=t^2$$
So we have two variables “time” and “distance” and they are related to each other.
Related to each value of the variable “t”, there is one value of the variable ‘d’.
Such a type of relationship between two variables is called a function. So the distance traveled by an object is a function of the time elapsed. And how the two variables are literally related, is represented by an equation as we saw earlier.
$$d=t^2$$
Now we are able to answer that what are functions in maths
This above equation tells us one specific relation between time and distance which is called a function in someway. But it’s just an example that we took. By using this example we will be able to find precise definition of function. So Let's see that definition:
Formal definition of function:
A function is a set of relationships between inputs to outputs where every input is assigned with a unique possible output.
Let’s say we have a notation of function
$$y=f(x)$$
Here,
x = Set of inputs from any set X
y = Set of outputs from set Y
So the definition of function is stated as:
“A function f from a set X to a set Y is a rule that assigns a unique (single) element f(x)∊Y to each element x∊X.”
So, i hope we will surely understand that what does function mean in calculus. It's just a relation which takes some input and process it into output values. These input and output values are callled as domain and range of function. Let's take an overview
Domain and Range
Well the function is relation between domain and range. It is the core concepts of functions in calculus. So let's see what is domain and range of function, it's importance and major examples to understand it.
Definition of Domain:
Domain? In simple words, the domain of a function is a set of all possible input values.
Again see the example of plant shadow and source light:
The inputs are source light and height of plant which are the domain of our function.
More formal,
$$y=f(x)$$
In a given notation of a function, every possible value of “x” is a domain for function y=f(x). because it gives a unique possible value of y.
Domain and Range Examples
We have a function,
$$y = x^2$$
In a given, your domain is x which gives a unique value of y every time. So, the domain of that function should be any value from [-∞,+∞], because we can put any value and it generates a result of unique value on each domain input.
Definition of Range:
The range is a set of all possible output values against the domain. Again, if we see an example of plant and source light, Whenever we change the direction of light or height of plant the output or range of our that function i.e shadow will change at every instance.
More formal,
$$y=f(x)$$
The range will be all possible values for y is there any good way to remember this how we remember that.
Let’s see again.
Domain and Range Examples
We have a function, $$y = x^2$$
Now what is the range of the function?
In a given function, the range is y which is a set of unique output results for our given domain like we have a domain of x2.
For a given situation, we have a domain that always gives a result of output value in positive as square transform negative inputting value to positive output value. So, the range will be [0,+∞].