Definition Formula And Solved Examples Of Composition And Inverse Functions
What is a composite function? Well, a composite function is usually composed of other functions such that the output of one function is the input of the other function. In other words, when the value of a function is found from two other given functions by applying one function to an independent variable and the other to the result of the other function whose domain consists of those values of the independent variable for which the result yielded by the first function lies in the domain of the second.
Example: Two functions - 3y+5 and y2 together forms a composite function which can be written as (3y+5)2
Explanation of Composition Functions
To form a composite function by a composition of two other functions we need to take two functions say g(x) = \[x^{2}\] and f(x) = x+5. Now, we need to put one function inside the other function so here we can put f(x) into g(x) to form a new function, called their composition.
As mentioned above, to form composite functions we need to insert one function into another. Here f(x) can be plugged into g(x) to form a function g(f(x)). We know that f(x) = x + 5, thus we can substitute the function in. Therefore, g(f(x)) = g(x + 5). Knowing the fact that g(x) = \[x^{2}\] we can insert the function and evaluate g(x + 5) = \[(x + 5)^{2}\]. Therefore, g(f(x)) = g(x + 5) = \[(x + 5)^{2}\] .
For practice, download composition of functions examples with answers pdf. By downloading composition of functions examples with answers pdf, you will have enough composite functions questions for practising.
Composite Functions Properties
There are four major properties of a composite function:
Property 1: Composite functions are not commutative
gof is not equal to fog
Property 2: Composite functions are associative
(fog)oh = fo(goh)
Property 3: A function f: A -B and g: B-C is one-one then gof: A-C is also one-one.
Property 4: A function f: A-B and g: B-C is onto then gof: A-C is also onto.
What is Inverse Function?
An inverse function is a function, which can reverse into another function. In other words, if any function “f” takes p to q then, the inverse of “f” i.e. “f-1” will take q to p. A function accepts a value followed by performing particular operations on these values to generate an output. If you consider functions, f and g are inverse, then f(g(x)) is equal to g(f(x)) which is equal to x.
Given below are the detailed summary of the Composition and inverse relation with examples:
Composite and Inverse Functions
Solved Examples
Question 1) Let f(x) = \[x^{2}\] and g(x) = \[\sqrt{1 - x^{2}}\] Find (gof)(x) and (fog)(x).
Solution 1) (gof)(x) = g(f(x)) = g(\[x^{2}\]) = \[\sqrt{1 - (x^{2})^{2} = \sqrt{1 - x^{4}}}\]
(fog) (x) = f(g(x)) = f (\[\sqrt{1 -x^{2})}\] = 1 -\[(x^{2})^{2}\] = 1 - \[x^{2}\]
Question 2) If f(x) =\[x^{2}\] , g(x) = \[\frac{x}{3}\] and h(x) = 3x+2 . Find out fohog(x).
Solution 2) h(g(x)) = 3 \[\left ( \frac{x}{3} \right )\] + 2 = x + 2
fohog(x) = f [h(g(x))] = \[(x + 2)^{2}\]
Therefore this is the required solution.
FAQs on Understanding Composition Of Functions And Inverse Of A Function
1. What is the composition of functions?
The composition of functions is a function formed by applying one function to the result of another function. If f and g are functions, then the composition is written as (f ∘ g)(x) = f(g(x)).
- First apply g to x.
- Then apply f to the result g(x).
2. How do you find the composition of two functions?
To find the composition of two functions, substitute one function into the other. Follow these steps:
- Step 1: Identify the inner function (e.g., g(x)).
- Step 2: Substitute g(x) into f(x).
- Step 3: Simplify the result.
3. Is composition of functions commutative?
No, composition of functions is not commutative, meaning (f ∘ g)(x) ≠ (g ∘ f)(x) in general. For example:
- Let f(x) = x + 1 and g(x) = x².
- (f ∘ g)(x) = f(x²) = x² + 1.
- (g ∘ f)(x) = g(x + 1) = (x + 1)² = x² + 2x + 1.
4. What is the domain of a composite function?
The domain of a composite function consists of all values of x for which the inner function is defined and its output lies in the domain of the outer function. To find it:
- Find the domain of g(x).
- Ensure g(x) is within the domain of f(x).
5. What is the inverse of a function?
The inverse of a function is a function that reverses the effect of the original function. If f(x) = y, then its inverse satisfies f⁻¹(y) = x. In other words, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The inverse undoes the original function.
6. How do you find the inverse of a function step by step?
To find the inverse function, interchange x and y and solve for y. Steps:
- Step 1: Let y = f(x).
- Step 2: Swap x and y.
- Step 3: Solve for y.
- Step 4: Replace y with f⁻¹(x).
- y = 2x + 3
- x = 2y + 3
- y = (x − 3)/2
7. What is the formula for the inverse of a linear function?
The inverse of a linear function f(x) = ax + b (a ≠ 0) is f⁻¹(x) = (x − b)/a. This is found by solving y = ax + b for x. For example, if f(x) = 5x − 7, then the inverse is f⁻¹(x) = (x + 7)/5.
8. How do you verify if two functions are inverses of each other?
Two functions are inverses if their compositions give x. That is, f(g(x)) = x and g(f(x)) = x. To verify:
- Find (f ∘ g)(x) and simplify.
- Find (g ∘ f)(x) and simplify.
- If both equal x, the functions are inverses.
9. What is the relationship between composition and inverse of a function?
The relationship is that a function composed with its inverse gives the identity function. Specifically, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This means the inverse reverses the effect of the original function, resulting in the identity function I(x) = x.
10. Can every function have an inverse?
No, a function has an inverse only if it is one-to-one (injective). A function is one-to-one if different inputs give different outputs. If a function fails the horizontal line test, it does not have an inverse unless its domain is restricted. For example, f(x) = x² is not one-to-one over all real numbers, but it has an inverse if defined for x ≥ 0.
