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A function that is a composite of two or three different functions is called a composite function. In other words, a composite function is a set of new functions which is created by the combination or composition of two different actions or functions. Invertible Functions or Inverse Function, as we go by the name, the inverse of a function means the opposite or reverse direction of a function. Meaning, if any function “f” takes p to q then, the inverse of “f” that is, “f-1” will take q to p. Moreover, a function accepts a value followed by performing particular operations on these values in order to generate an output. Here, we will be discussing the composite function and invertible function along with some examples and important questions.
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Keyterms: Function, Composite function, Invertible Functions, Inverse Function, Relation, Variable function
Read more: Circumference of Circle
What is Composite Function?
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Composite function refers to the resultant value of two specified functions. When the output derived from the application of a function with a second independent variable function becomes the input of the third function, then it is called a composite function. Also, whose scope includes the values of the independent variable for which the result of the first function is placed in the domain of the second.
In Mathematics, the composition of a function is a process, where two functions say f and g create a new function say h in such a way that h (x) = g (f (x)). Here, we can see function g applies to the function of x i.e., f (x)
Let f: A → B and g: B → C are two functions.
So, the composition of f and g, denoted by gof, is known as the function:
g of: A → C given by gof (x) = g (f (x)), A x ∈ A.
Also check: Properties of Inverse Trigonometric Functions
Importance of Composite Functions
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Both concepts Composite and Inverse functions have a real-life application like in industries, hospitals, etc. They work in a process where many important functions are interlinked. The inverse of a function and the composition of a function are two mathematical notions having practical applications. The aim of the composition of functions and inverse of a function is to develop application-based thinking of how the functions work.
Relation and Function
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To represent the relation and function of a composite function, some symbols and domains are used, which are given below:
Symbol:- The composite function g (f (x)) is also written as g o f, which specifies the relation between functions g and f. Where o is the composition operator which is used to define the composition of functions.
Domain:- g (f (x)) is read as g of f of x or “the composition g and f”. In the composition of
(g of) (x) the domain of function g becomes f (x). The domain is a collection of all the values that make up the function.
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Properties of Composite Functions
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- The composition of functions such as f, g, and h is composable because there is some relation or association between functions. The composite function always has associative property. If f º (g º h) = (f º g) º h.
- In special circumstances, some particular functions have commutativity a special property, For example, |x| + 3 = |x + 3| only when x ≥ 0. The functions g and f are said to commute with each other if g º f = f º g.
- The composition of one-to-one (injective) functions is always one-to-one.
- Likewise, onto (surjective) function composition is always onto.
- The inverse composition of the functions f and g are equal to the inverse composition of both functions., like (f º g) −1 = g−1º f−1.
Explanation of Composite Functions
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Now we know composite functions are those that have a function placed inside another function.
Let’s suppose we have two functions - say f (x) = x2 and g (x) = 3x
By combining two functions, we can create a new function called their composition.
Let’s see what happens when we are trying to put g (x) inside f (x).
Instead of putting an x in the function f (x), we’ll put g (x) in there and write it as f (g (x)).
When you’re combining functions, you should always remember to work from the inside out.
Since we know that g (x) = 3x, we can substitute that in.
Therefore, f (g (x)) = f (3x).
To finish the composition, we use the fact that f (x) = x2
To evaluate f (3x) = 3x2
Now, we’re done! Therefore, f (g (x)) = f (3x) = (3x) 2 = 9x2
Also check: Addition of Vectors
What is Invertible Function?
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Invertible Functions or Inverse Function, as we go by the name, the inverse of a function means the opposite or reverse direction of a function. To solve an inverse of a function, we need to simply put the process of a function in an opposite direction.
For Example: -
Details of Inverse of a function f (x) = 4x + 3, shown as a flow diagram: -
So, the inverse of f (x) = 4x+3 is f-1 (x) = x-3
4
The inverse is shown by putting “-1” after the function name
Like, f-1 (x) and it is pronounced as “f inverse of x”
So, the inverse of f (x) = 4x+3 is written as f-1 (x) = x-¾
Also read: Surface area and volume notes
Explanation of Inverse Function
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An inverse function, which is written f−1 (x) f−1 (x), is defined as the inverse function of f (x) f (x) if it completely reverses the f (x) f (x) process. If f (x) f (x) turns aa into bb, then f-1 (x) f-1 (x) must turn bb into aa. The function
f-1 (x) f-1 (x) is the inverse function of f (x) f (x) if:
f (f-1(x))=x
One important feature of the inverse function is that it gives us back the original result.
If we take the same example of water,
To hold this statement true, let’s take the above example f (x) = 4x+3
Let’s assume x = 2,
Then, f (x)= 4x+3, f (2) = 4*2+3= 11
We can now use inverse of 11,
f-1(x)= x-3/4, f-1 (11)= 11-3/4= 2
by doing inverse, we get the value of x=2
i.e. f-1 (f (2)) = 2, we can say “f inverse of f of 2 is 2”
So, applying a function f and then its inverse f-1gives us the original value back again:
f-1 (f(x) ) = x
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Things to Remember
- A composite function is a set of new functions which is created by the combination or composition of two different actions or functions.
- In Mathematics, the composition of a function is a process, where two functions say f and g create a new function say h in such a way that h (x) = g (f (x)). Here, we can see function g applies to the function of x i.e., f (x)
- The composition of f and g, denoted by gof, is defined as the function:
- g of: A → C given by gof (x) = g (f (x)), ∀ x ∈ A.
- The aim of the composition of functions and inverse of a function is to develop application-based thinking of how the functions work.
- The composite function always has associative property. If f º (g º h) = (f º g) º h.
- The inverse of a function means the opposite or reverse direction of a function.
- One important feature of the inverse function is that it gives us back the original result.
Also read: Circles Revision Notes
Sample Questions
Ques: f (x)=x2 - x + 3 g (x)= 2x - 1. Solve (a) (fog) (x) (b) (gof) (x). (4 marks)
Ans: We have been given two functions f (x) and g (x). These are composition function problems. Composition functions are defined as functions whose result is got from the results of another function.
- (fog) (x)
This is a composition function. We read it as f of g. We have to substitute g into the function f and get the result.
(fog) (x)=f(g(x))=f(2x-1)=(2x-1)2-(2x-1)+3=4x2-4x+1-2x+1+3=4x2-6x+5
- (gof) (x)
This is a composition function. We read it as g of f. We have to substitute f into the function g and get the result.
(gof) (x)=g (f (x)) =g (x2-x+3) =2 (x2-x+3)-1=2x2-2x+6-1=2x2-2x+5
Ques: If f: A → B, f (x) = y = x2 and g: B→C, g (y) = z = y + 2 find g o f. Given A = {1, 2, 3, 4, 5}, B = {1, 4, 9, 16, 25}, C = {2, 6, 11, 18, 27}. (2 Marks)
Ans: g o f(x) = g(f(x))
g(f(1)) = g(1) = 2, g(f (2)) = g(4) = 6, g(f(3)) = g(9) = 11, g(f(4)) = g(16) = 18, g(f(5)) = g(25) = 27.
Ques: If there are three functions, such as f (x) = x, g (x) = 2x and h (x) = 3x. Then find the composition of these functions, such as [f º (g º h)] (x) for x = -1. (3 marks)
Ans: Given,
f (x) = x, g(x) = 2x
h (x) = 3x
To find: [f º (g º h)] (x)
[f º (g º h)] (x) = f º (g (h(x)))
= f º g (3x)= f(2(3x))= f(6x)= 6x
If x = -1, then;
[f º (g º h)] (-1) = 6 (-1) = -6
Ques: Write the inverse of the above g o f. (2 marks)
Ans: (g o f) -1 = f-1(g-1(z))
f-1 (g-1(z)) = f-1(g-1(2)) = f-1(1) = 1, f-1(g-1(6)) = f-1(4) = 2, f-1(g-1(11)) = f-1(9) = 3, f-1(g-1(18)) = f-1(16) = 4
and f -1 (g-1 (27)) = f-1 (25) = 5.
Ques: What functions are not invertible? (2 marks)
Ans: Functions are non-invertible because when taking the inverse, the graph becomes a parabola that opens to the right that is not a function. A sideway opening parabola comprises two outputs for every input that is not a function by definition. Furthermore, by reducing the domain, you can make the function invertible.
Ques: If f(x) = x2, g(x) = x/3 and h(x) = 3x+2 . then find out fohog(x). (2 marks)
Ans: h(g(x)) = 3 (x/3) + 2 = x + 2
fohog(x) = f
h(g(x)) = (x + 2)2.
Ques: Let f(x) = x2 and g(x) = √1 - x2, then find (gof)(x) and (fog)(x). (2 marks)
Ans: (gof)(x) = g(f(x)) = g(x)2 = √1 - (x2)2 = √1 - x4
(fog) (x) = f(g(x)) = f (√1 - x2) = 1 - (x2)2 = 1 - x2
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