Definition Formula Properties And Solved Examples Of Composition Of Functions And Inverse Functions
Composition of a function and its inverse are two mathematical concepts with practical applicability. The objective of these two concepts is to increase the understanding of functions and all the terms related to it. Students will better understand the definitions of domain and range after going through these two concepts.
The objective of the composition of functions and inverse of a function is to develop an application based thinking of how the functions work. Both of these concepts have a real-life application. Students are advised to regularly give time and effort to mathematics and increase their score in it.
Composite Functions and Inverse Functions
Let us try and understand both of these mathematical concepts in detail. Composition of functions and inverse functions are easy concepts to understand and apply. Students are advised to try as many examples as possible to solidify their learning and understanding of both the concepts. Given below is the detailed explanation of both the concepts:
Composition of Functions
Composition of function is defined when the result of a function is obtained by applying another function. The independent variable is another function. Let us try to understand the composition of functions with the help of an example.
Let there be two functions, f(x) and g(x).
f(x) = 2x + 1 g(x)=x2
Let us find the value of g(x) with the result obtained from f(x).
Calculate f(x) at 1.
f(1)=2.1+1=3
Let us calculate g(x) at 3.
g(3)=32= 9.
To streamline the above process and understand it better, we create a new function. This is how we represent the composition of functions.
f(g(x))=(fog)(x)
Where o is the composition operator and is used to define the composition of functions. Let us try and solve the above problem with this representation at x=1.
(gof)(x)=g(f(x))
=g(2x+1)
=(2x+1)2
=4x2+ 4x + 1
= 9
The notation (fog) is read as f of g or f is composed of g.
Inverse Functions
Inverse functions, as the name suggests, is to describe an inverse relationship between two functions. The two functions are opposite of each other. Let us try and understand this concept using a common example.
Let us take the case of temperature scales. Two scales, degree Celsius scale and the Fahrenheit scale, are used to measuring temperature.
C(x)=5/9(x-32) converts fahrenheit to degree celsius.
F(x)=9/5(x) + 32 converts degree celsius into degree fahrenheit.
We wish to convert 77oF into degree celsius.
C(x)=5/9(77-32)
=5/9(45)
=5.5
=25oC
Now to understand the inverse function, let us convert this degree celsius into degree fahrenheit.
F(x)=9/5(x)+32
=9/5(25)+32
=9.5+32
=45+32
=77oF
We get the same Fahrenheit value we began with. Inverse functions are opposite of each other. We can obtain any of the functions.
C(x) = 5/9(x-32)
9/5(C(x)) = x-32
9/5(x) + 32 = F(x)
This is how the inverse functions work.
FAQs on Introduction To The Composition Of Functions And Inverse Of A Function In Algebra
1. What is the composition of functions?
The composition of functions is a new function formed by applying one function to the result of another function. It is written as (f ∘ g)(x) = f(g(x)).
In composition:
- The function g(x) is applied first.
- The output of g(x) becomes the input of f(x).
- The result is a single combined function.
2. How do you find the composition of two functions?
To find the composition of two functions, substitute one function into the other. Use the formula (f ∘ g)(x) = f(g(x)).
Steps:
- Identify the inner function (g).
- Substitute g(x) into f(x).
- Simplify the expression.
3. What is the formula for inverse of a function?
The inverse of a function is found by interchanging x and y and solving for y, and it is denoted by f⁻¹(x).
Steps to find f⁻¹(x):
- Write y = f(x).
- Interchange x and y.
- Solve for y.
- Replace y with f⁻¹(x).
4. How do you check if two functions are inverses of each other?
Two functions are inverses if their composition equals the identity function, meaning (f ∘ g)(x) = x and (g ∘ f)(x) = x.
Steps:
- Find (f ∘ g)(x).
- Find (g ∘ f)(x).
- If both results equal x, the functions are inverses.
5. What is the difference between composition of functions and inverse of a function?
The composition of functions combines two functions into one, while the inverse of a function reverses the effect of a single function.
- Composition: (f ∘ g)(x) = f(g(x)) combines operations.
- Inverse: f⁻¹(x) undoes what f(x) does.
- If f and g are inverses, then (f ∘ g)(x) = x.
6. Does the order matter in composition of functions?
Yes, the order matters because (f ∘ g)(x) ≠ (g ∘ f)(x) in general.
Example:
- Let f(x) = x + 1 and g(x) = x².
- (f ∘ g)(x) = x² + 1.
- (g ∘ f)(x) = (x + 1)² = x² + 2x + 1.
7. What are the conditions for a function to have an inverse?
A function has an inverse if it is one-to-one (injective), meaning each output corresponds to exactly one input.
- No two different x-values give the same y-value.
- It must pass the horizontal line test.
- Its domain and range swap in the inverse function.
8. How do you find the inverse of a quadratic function?
To find the inverse of a quadratic function, restrict its domain and then interchange x and y and solve for y.
Example: Let f(x) = x² with domain x ≥ 0.
- Write y = x².
- Interchange: x = y².
- Solve: y = √x.
9. What happens to the domain and range in inverse functions?
In inverse functions, the domain and range interchange.
- If f has domain A and range B,
- Then f⁻¹ has domain B and range A.
10. Can you give a real-life example of composition of functions?
A real-life example of composition of functions is calculating total cost after tax is added to a discounted price.
Suppose:
- g(x) gives discounted price: g(x) = 0.8x (20% discount).
- f(x) adds tax: f(x) = 1.1x (10% tax).
