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Composite and inverse functions are fundamental concepts in mathematics, especially in calculus and algebra. Understanding these functions is important for solving complex mathematical problems. A composite function is created by combining two functions, where the output of one function becomes the input of another. An inverse function, on the other hand, reverses the effect of the original function, mapping the output back to the input. These functions play a significant role in various mathematical applications, such as function transformations, solving equations, and real-world modeling. Key properties such as domain, range, associativity, and bijectivity distinguish these functions. Competitive exams like JEE (Joint Entrance Examination), NEET (National Eligibility cum Entrance Test) and GATE frequently test these concepts through various problems, making it important for students to master these topics.
Table of Contents
What are Composite Functions?
A composite function is a function that is formed by combining two functions such that the output of one function becomes the input of another. Mathematically, if there are two functions, say f(x) and g(x), the composite function of f and g, denoted by (f∘g)(x) or f(g(x)), means applying g(x) first and then applying f(x) to the result. In simple terms, composite functions allow you to “chain” functions together to produce a new function.
Important Terms:
- Domain: The set of input values for which the composite function is defined. The domain of the composite function is restricted by both the domain of the inner function g(x) and the domain of the outer function f(x).
- Range: The set of possible output values that result from applying the composite function. This depends on the range of the inner function and how it fits into the domain of the outer function.
- Inner Function (g(x)): The function that is applied first in the composition. It provides the input for the outer function.
- Outer Function (f(x)): The function applied after the inner function. It takes the output of the inner function and processes it further.
- Function Composition: The process of combining two or more functions into a single function, where the output of one function becomes the input for the next.
Example:
Let f(x) = 2x+3 and g(x) = x2. The composite function (f∘g)(x) is calculated as:
(f∘g)(x) = f(g(x)) = f(x2) = 2(x2)+3 = 2x2+3
Important Points:
- A composite function must satisfy the condition that the range of the inner function is within the domain of the outer function.
- Composite functions are often used in calculus and algebra to simplify complex expressions and functions.
Must Read How to Solve Equations With Variables on Both Sides?
Properties of Composite Functions
Composite functions possess several important properties that help in their manipulation and application. Understanding these properties is crucial when solving problems involving function composition. Here are the key properties:
1. Associativity of Function Composition
Function composition is associative, meaning that when you compose multiple functions, the order of composition doesn’t matter as long as the sequence remains the same. That is, for three functions f(x), g(x), and h(x):
(f∘(g∘h))(x) = ((f∘g)∘h)(x)
This property ensures that you can group the functions in any way without changing the result.
Example: Let f(x) = 2x, g(x) = x+3, and h(x) = x2.
- g(h(x)) = g(x2) = x2+3
- Now, f(g(h(x))) = f(x2+3) = 2(x2+3) = 2x2+6
- Alternatively, (f∘g)(x) = f(g(x)) = f(x+3) = 2(x+3) = 2x+6, and then (f∘g)(h(x)) = 2(x2)+6 = 2x2+6.
In both cases, the result is the same, illustrating the associativity of composition.
2. Not Commutative
In general, function composition is not commutative, meaning f∘g ≠ g∘f for most functions. The order in which you compose the functions affects the result. For example, if f(x) = x+1 and g(x) = 2x, then:
(f∘g)(x) = f(g(x)) = 2x+1 and (g∘f)(x) = g(f(x)) = 2(x+1) = 2x+2
Example: Let f(x) = x2 and g(x) = x+2.
- (f∘g)(x) = f(g(x)) = (x+2)2 = x2+4x+4
- (g∘f)(x) = g(f(x)) = g(x2) = x2+2
The two compositions f∘g and g∘f yield different results, demonstrating that composition is not commutative.
3. Identity Function
The composition of any function f(x) with the identity function I(x) = x results in the function itself. That is:
f∘I = f and I∘f = f
The identity function acts as a “neutral element” in function composition, leaving the original function unchanged.
Example: Let f(x) = 3x+1 and the identity function I(x) = x.
- f(I(x)) = f(x) = 3x+1
- I(f(x)) = f(x) = 3x+1
The function remains unchanged when composed with the identity function.
4. Domain and Range
The domain of a composite function (f∘g)(x) is restricted to the set of values for which the inner function g(x) is defined and its output falls within the domain of the outer function f(x). Formally:
Domain of (f∘g)(x) = {x ∈ Domain of g∣g(x) ∈ Domain of f}
The range of the composite function depends on both the range of the inner function and how it interacts with the outer function.
Example: Let f(x) = √x and g(x)=x−2.
- The domain of g(x) = x−2g(x) = x – 2 is all real numbers (−∞,∞).
- The domain of f(x) = √x is [0,∞).
- For the composite function (f∘g)(x) = √(x−2), the domain is x−2 ≥ 0, or x ≥ 2.
5. Inverse of a Composite Function
The inverse of a composite function f∘g, if it exists, is given by reversing the order of composition and finding the inverse of each function. Specifically:
(f∘g)-1 = g-1∘f-1
This property holds only when both functions f and g are invertible.
Example: Let f(x) = 2x and g(x) = x+3.
- The inverse of f(x) = 2x is f-1(x) = x/2.
- The inverse of g(x) = x+3 is g-1(x) = x−3.
- The inverse of the composite function (f∘g)(x) = 2(x+3) = 2x+6 is:
(f∘g)-1 (x) = g-1(f−1(x)) = g−1(x/2) = (x/2)−3
6. Continuity and Differentiability
If both functions f and g are continuous, then their composite function (f∘g)(x) is also continuous. Similarly, if both functions are differentiable, the composite function is differentiable, and its derivative is given by the chain rule:
d/dx (f(g(x))) = f′(g(x))⋅g′(x)
Example (Continuity): Let f(x) = 2x and g(x) = x2. Both functions are continuous, so their composite (f∘g)(x) = 2x2 is also continuous.
Example (Differentiability): Let f(x) = sin(x) and g(x) = x3. Both functions are differentiable. By the chain rule:
d/dx sin(x3) = cos(x3)⋅3x2
What are Inverse Functions?
An inverse function essentially “undoes” the action of a given function. If a function f maps an element x from its domain to an element y in its range, then the inverse function f-1 maps y back to x. In other words, for a function f(x), its inverse f-1 satisfies the following relationships:
f(f-1(x)) = x and f-1(f(x)) = x
This means that applying the inverse function to the output of the original function returns the input.
Definition: If f(x) is a function from the set A to B, then its inverse f-1 is a function from B back to A such that for every y ∈ B:
f-1(y) = x if f(x) =y
The function f-1 exists only if fff is a bijective function (both one-to-one and onto).
Example:
Let f(x) = 2x+3.
- To find the inverse, we solve for x in terms of y:
Y = 2x+3 ⇒ x = (y−3)/2
- So, the inverse function is:
f-1(x) = (x−3)/2
Verification:
- f(f-1(x)) = f((x−3)/2) = 2((x−3)/2)+3 = x
- f-1(f(x)) = f-1(2x+3) = ((2x+3)−3)/2 = x
Properties of Inverse Functions
Inverse functions possess several unique properties that are important for solving various mathematical problems. Here are the important properties of inverse functions along with examples to illustrate them:
1. Reversibility
The most important property of inverse functions is that they “reverse” each other. If f(x)) and f-1(x) are inverse functions, then for all x in the domain of f and f-1:
f(f-1(x)) = x and f-1(f(x)) = x
Example: Let f(x) =2x+3. The inverse of this function is f-1(x) = (x−3)/2.
- f(f-1(x)) = f((x−3)/2) = 2((x−3)/2)+3 = x
- f-1(f(x)) = f-1(2x+3) = ((2x+3)−3)/2 = x
Both operations return x, demonstrating reversibility.
2. Domain and Range Switching
The domain of a function f(x) becomes the range of its inverse f-1(x), and the range of f(x) becomes the domain of f-1(x). If f : A→B, then f-1 : B→A.
Example: Let f(x) = √x with domain [0,∞) and range [0,∞).
The inverse of f(x) = √x is f-1(x) = x2, which has a domain of [0,∞) and a range of [0,∞).
- Domain of f(x): [0,∞)
- Range of f(x): [0,∞)
- Domain of f-1(x): [0,∞)
- Range of f-1(x): [0,∞)
The domain and range of the original function and its inverse are swapped.
3. Bijectivity (One-to-One and Onto)
A function must be bijective (both injective and surjective) to have an inverse. An injective function (one-to-one) ensures that different inputs map to different outputs, and a surjective function (onto) ensures that every element in the codomain has a corresponding preimage in the domain.
Example: Let f(x) =3x+5.
- Injective: For f(x) = 3x+5, if f(x1) = f(x2), then 3x1+5 = 3x2+5, which simplifies to x1 = x2. Thus, f(x) is injective.
- Surjective: For every y ∈ R, there exists an x ∈ R such that f(x) = y. Solving y = 3x+5 for x, we get x = (y−5)/3, which is always possible. Therefore, f(x) is surjective.
Since f(x) is both injective and surjective, it has an inverse f-1(x) = (x−5)/3.
4. Symmetry of Graphs
The graphs of a function f(x) and its inverse f-1(x) are symmetric with respect to the line y=x. This is because the inverse swaps the roles of x and y.
Example: Let f(x) = x2 (restricted to x ≥ 0) and its inverse f-1(x) = √x.
- The graph of f(x) = x2 is a parabola.
- The graph of f-1(x) = √x is half of a sideways parabola.
If you reflect the graph of f(x) over the line y = x, you get the graph of f-1(x).
5. Inverse of a Composite Function
The inverse of a composite function f∘g is the composition of the inverses in the reverse order:
(f∘g)-1 = g-1∘f-1
This property holds if both f and g are invertible.
Example: Let f(x) = 2x and g(x) = x+3.
- The inverse of f(x) = 2x is f-1(x) = x/2.
- The inverse of g(x) = x+3 is g-1(x) = x−3.
The composite function (f∘g)(x) = f(g(x)) = 2(x+3) = 2x+6.
The inverse of f∘g is:
(f∘g)-1(x) = g-1(f-1(x)) = ((x/2)−3)
6. Derivatives of Inverse Functions (Inverse Function Theorem)
If f(x) is differentiable and invertible, and f-1(x) is also differentiable, then the derivative of the inverse function is given by:
(d/dx)[f-1(x)] = 1/(f′(f-1(x)))
This formula helps to find the derivative of the inverse function without explicitly solving for it.
Example: Let f(x)=x3, which is invertible. The inverse is f-1(x) = 3√x.
- The derivative of f(x) is f′(x) = 3x2.
- According to the inverse function theorem:
d/dx[f-1(x)] = 1/(3(f−1(x)))2 = 1/3(3√x2)
7. Inverse of Trigonometric Functions
The inverse of trigonometric functions such as sine, cosine, and tangent can be defined for specific domains where the function is bijective. These inverses are called arcsine, arccosine, and arctangent.
Example: The sine function sin(x) is not invertible over its entire domain, but it is invertible when restricted to [−π/2,π/2].
- The inverse of sin(x) in this interval is sin-1(x), or arcsin x.
- For example, sin-1(1) = π/2, meaning sin(π/2) = 1.
Difference between Composite and Inverse Functions
Here is a comparison between Composite and Inverse Functions.
| Particular | Composite Functions | Inverse Functions |
| Definition | A function formed by applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). | A function that “reverses” another function, denoted as f-1(x), such that f(f-1(x)) = x. |
| Purpose | Combines two or more functions into one, applying them in sequence. | Undoes the effect of the original function, returning to the original input. |
| Expression | (f∘g)(x) = f(g(x)) | f-1(x) |
| Existence Conditions | Exists for any two functions f(x) and g(x) that can be applied in succession. | Exists only if the function is bijective (one-to-one and onto). |
| Order of Operation | The first function is applied after the second: f(g(x)). | The inverse function undoes the original function: f-1(f(x)) = x. |
| Graphical Interpretation | The graph of f(g(x)) combines the transformations of both f(x) and g(x). | The graph of f-1(x) is a reflection of f(x) across the line y = x. |
| Commutativity | Composite functions are not generally commutative: f(g(x)) ≠ g(f(x)). | The inverse undoes the original, so f(f-1(x)) = f-1(f(x)) = x. |
| Derivative (for differentiable functions) | The derivative of a composite function is found using the chain rule: d/dx [f(g(x))] = f′(g(x))⋅g′(x). | The derivative of an inverse function is d/dx [f-1(x)] = 1/f′(f-1(x)). |
| Example | f(x) = x2, g(x) = 2x → (f∘g)(x) = (2x)2 = 4x2. | f(x) = 2x+3, inverse f-1(x) = (x−3)/2. |
| Symmetry | No specific symmetry for composite functions. | The graph of a function and its inverse are symmetric across the line y=x. |
| Application | Used to simplify complex operations by combining functions. | Used to reverse processes, often in solving equations and inverting transformations. |
Solved Examples of Composite and Inverse Functions
Here are 5 solved examples, including composite and inverse functions, to help clarify their concepts.
Composite Functions:
Example 1:
Given:
f(x) = 2x+1
g(x) = x2
Find (f∘g)(x) and (g∘f)(x).
Solution:
- (f∘g)(x) means applying g(x) first, then f(x):
(f∘g)(x) = f(g(x)) = f(x2) = 2(x2)+1 = 2x2+1 - (g∘f)(x) means applying f(x) first, then g(x):
(g∘f)(x) = g(f(x)) = g(2x+1) = (2x+1)2 = 4x2+4x+1
Example 2:
Given:
f(x) = sin(x)
g(x) = x+π/2
Find (f∘g)(x).
Solution:
- (f∘g)(x) means applying g(x) first, then f(x):
(f∘g)(x) = f(g(x)) = f(x+π/2) = sin(x+π/2)
Using the trigonometric identity sin(x+π/2) = cos(x)
Example 3:
Given:
f(x) = 3x−2
g(x) = x/3
Find (f∘g)(x) and (g∘f)(x).
Solution:
- (f∘g)(x) means applying g(x) first:
(f∘g)(x) = f(g(x)) = f(x/3) = 3(x/3)−2 = x−2 - (g∘f)(x) means applying f(x) first:
(g∘f)(x) = g(f(x)) = g(3x−2) = (3x−2)/3 = x−(⅔)
Inverse Functions:
Example 4:
Given:
f(x)= 3x−7
Find the inverse function f-1(x).
Solution:
To find the inverse, solve for x in terms of y:
y = 3x−7 ⇒ y+7 = 3x ⇒ x = (y+7)/3
Thus, the inverse function is:
f-1(x) = (x+7)/3
Verification:
- f(f-1(x)) = 3((x+7)/3)−7 = x
- f-1(f(x)) = ((3x−7)+7)/3 = x
Example 5:
Given:
f(x) = (x−1)/(x+2)
Find the inverse function f-1(x).
Solution:
- Solve y = (x−1)/(x+2) for x:
y(x+2) = x−1 ⇒ yx+2y = x−1
yx−x = −1−2y ⇒ x(y−1) = −2y−1 ⇒ x = (−2y−1)/y−1
Thus, the inverse function is:
f-1(x) = (−2x−1)/(x−1)
FAQs
Inverse functions are functions that “undo” each other. For example, if f(x) = 2x + 1, then its inverse function would be f⁻¹(x) = (x – 1)/2. Composite functions are functions formed by applying one function after another. For example, if f(x) = x² and g(x) = 3x, then the composite function f(g(x)) would be (3x)².
A composite function is a function formed by applying one function to the result of another function. For example, if f(x) = x² and g(x) = 2x + 1, then the composite function f(g(x)) is equal to (2x + 1)².
A composite function is a function formed by applying one function to the result of another function. It’s called “composite” because it’s created by combining two or more functions into a single expression.
The inverse of a function is another function that undoes the original function. If you apply a function and then its inverse, you end up back where you started.
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