Composite Functions
When two functions are combined in such a way that the output of one function becomes the input to another function, then this is referred to as composite function.
Consider three sets X, Y and Z and let f : X → Y  and g: Y → Z.
According to this, under map f, an element  x ∈ X is mapped to an element y = f(x) ∈ Y which in turn is mapped by g to an element z ∈ Z  in such a manner that z = g(y) = g[(f(x)] .
This mapping comprising of mappings f and g is known as composition of mappings. It is denoted by gof . Therefore, we are mapping onto .
The composite function is denoted as:
\(\begin{array}{l}~~~~~~~~~\end{array} \)
( gof)(x) = g(f (X) )
Similarly, (fog) (x) = f (g(x))
So, to find (gof) (x), take f(x) Â as argument for the function g.
Learn more about composition of functions here.
Let us try to solve some questions based on composite functions.
| Let’s Work Out:
Example 1: Given the function f(x) = 3x + 5 Â and g(x) = \(\begin{array}{l} 2x^3 \end{array} \) Â .Find ( gof)(x) and ( fog)(x).
Solution: Â We know, (gof) (x) = g(f(x)) = g (3x+5) = \(\begin{array}{l} 2(3x+5)^3 \end{array} \)
Using Binomial Expansion, we have
\(\begin{array}{l}(gof) (x) = 2 \left [ (3x)^{3} + 3.(3x)^{2}.5 + 3.(3x)(5)^{2}+(5)^3 \right ]\end{array} \)
\(\begin{array}{l}\Rightarrow (gof) (x) = 2 \left [ 27x^{3} + 135x^{2} + 225 x + 125 \right]\end{array} \)
\(\begin{array}{l}\Rightarrow (gof) (x) = 54x^{3} + 270x^{2} + 450 x + 250 \end{array} \)
Now, (fog)(x) = f \(\begin{array}{l}\left( g(x) \right)\end{array} \) = f ( \(\begin{array}{l}2x^3\end{array} \) ) = \(\begin{array}{l}3 (2x^{3}) + 5 \end{array} \)
\(\begin{array}{l}\Rightarrow (fog)(x) = 6x^{3} + 5\end{array} \)
Example 2: Let f(x) = \(\begin{array}{l}x^2\end{array} \) and g(x) = \(\begin{array}{l} \sqrt{1 – x^2}\end{array} \) .Find (gof)(x) and ( fog)(x) .
Solution: (gof)(x) = g( f(x)) = g(\(\begin{array}{l}x^2\end{array} \) ) = \(\begin{array}{l} \sqrt{1 – (x^2)^2}\end{array} \) = \(\begin{array}{l} \sqrt{1 – x^4} \end{array} \)
(fog) (x) = f(g(x)) = f( \(\begin{array}{l} \sqrt{1 – x^2} \end{array} \) ) = ( \(\begin{array}{l} \sqrt{1 – x^2})^2\end{array} \) = \(\begin{array}{l} 1 – x^2 \end{array} \) |
Inverse of a Function
Let f:X → Y. Now, let f represent a one to one function and y be any element of Y , there exists a unique element x ∈ X  such that y = f(x).Then the map
\(\begin{array}{l}~~~~~~~~~~\end{array} \)
\(\begin{array}{l} f^{-1}:f[X] \rightarrow X \end{array} \)
which associates to each element is called as the inverse map of f.
The function f(x) = \(\begin{array}{l} x^5 \end{array} \)
and g(x) = \(\begin{array}{l} x^{\frac 12} \end{array} \)
 have the following property:
\(\begin{array}{l} f\left( g(x) \right) \end{array} \)
= \(\begin{array}{l} f \left( x^{\frac 15} \right) \end{array} \)
= \(\begin{array}{l} (x^{\frac 15} )^5 \end{array} \)
= x
\(\begin{array}{l} g\left( f(x) \right) \end{array} \)
= \(\begin{array}{l} g \left( x^{5} \right) \end{array} \)
= \(\begin{array}{l} (x^{5} )^{\frac 15} \end{array} \)
= x
Thus, if two functions f and g  satisfy \(\begin{array}{l} f \left( g(x) \right) \end{array} \)
= x for every x in domain of f , then in such a situation we can say that the function f is the inverse of g  and g  is the inverse of f .
For finding the inverse of a function, we write down the function y as a function of x  i.e. y = f(x)  and then solve for x  as a function of y.
To have a better insight on the topic let us go through some examples.
| Let’s Work Out:
Example: 1 If f(x) = \(\begin{array}{l}x^2\end{array} \) , g(x) = \(\begin{array}{l} \frac{x}{3} \end{array} \) Â Â and h(x) = 3x+2 . Find out fohog(x).
Solution: Â \(\begin{array}{l} h\left( g(x) \right) \end{array} \) = 3 \(\begin{array}{l} \left ( \frac x3 \right) \end{array} \) + 2 = x + 2
fohog(x) = f \(\begin{array}{l} \left[ h\left(g(x)\right)\right] \end{array} \) = \(\begin{array}{l} ( x+ 2)^2 \end{array} \)
This is the required solution.
Example 2: Find the inverse of the function f(x) = \(\begin{array}{l} x^3 \end{array} \) , x ∈ R.
Solution: The given function f(x) = \(\begin{array}{l} x^3 \end{array} \)   is a one to one and onto function defined in the range → R  . Therefore, we can find the inverse of this function.
To find the inverse, we need to write down this function as
\(\begin{array}{l}~~~~~~~~~~~~~\end{array} \) y = \(\begin{array}{l} x^3 \end{array} \)
In the above equation,y  is an arbitrary element from the range of f. If we solve for x from the above equation, we will get:
\(\begin{array}{l}~~~~~~~~~~~~~\end{array} \) x = \(\begin{array}{l} y^{\frac 13} \end{array} \)
This gives a function g:Y →X. This new function g can be defined as
\(\begin{array}{l}~~~~~~~~~~~~~\end{array} \) g(y) = \(\begin{array}{l} y^{\frac 13} \end{array} \)
This function g is the inverse of the function f since it’s domain is same as the range of the function f.Since g(y) = \(\begin{array}{l} y ^{\frac 12} \end{array} \) , representing the independent variable with x , we get g(x) = \(\begin{array}{l} x^{\frac 13}\end{array} \) = \(\begin{array}{l} f^{-1}(x) \end{array} \) .. |
Practice Problems
- Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as f = {(1, 1), (2, 2), (3, 3)}. Find f–1, if it exists.
- Find gof and fog, if f(x) = |x| and g(x) = |5x – 2|.
- Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
So, wasn’t that easy and fun? We have learnt about composition of functions and inverse functions. To learn more about functions, please visit our website www.byjus.com or download our app-BYJU’s the learning app. Happy Learning!!
Frequently Asked Questions – FAQs
Q1
What are the conditions for the inverse function?
The conditions for two functions f and g to be inverses:
f(g(x)) =x for all x in the domain of g
g(f(x)) = x for all x in the domain of f
If f and g are inverses, composing f and g (in either order) creates the function that returns that input called the identity function for every input.
Q2
What is the difference between inverse function and composite function?
A composite function is a function obtained when two functions are combined so that the output of one function becomes the input to another function.
A function f: X → Y is defined as invertible if a function g: Y → X exists such that gof = I_X and fog = I_Y. The function g is called the inverse of f and is denoted by f ^–1.
Q3
What is the symbol of an inverse function?
The symbol of an inverse function is -1. That means f^-1 denotes the inverse of function f.
Q4
Does every function have an inverse?
No, not every function has an inverse. If a function f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.
Q5
Why does a function have no inverse?
A function has no inverse if it does not follow the conditions to be invertible.
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