Composition of Functions and Invertible Function

Introduction

Composition of functions and inverse functions are fundamental concepts in mathematics that play a crucial role in solving equations, analyzing relationships, and constructing mathematical models. Understanding these concepts is essential for students to grasp the power and flexibility of functions.

Composition of Functions

The composition of two functions $f$ and $g$ , denoted by $f\circ g$, is a new function that combines the operations of $f$ and $g$. It is defined as follows:

$(f\circ g)(x)=f(g(x))$

In other words, to evaluate $f\circ g$ at a value $x$, we first evaluate $g(x)$ and then use the result as the input for $f$.

Properties of Composition

1. Associativity: $(f\circ g)\circ h=f\circ (g\circ h)$
2. Identity Function: $f\circ i=f=i\circ f$ where $i$ represents the identity function $(i(x)=x)$

Examples of Composition

1. Suppose $f(x)=x^{2}$ and $g(x)=2x+1$, Then, $(f\circ g)(x)=f(g(x))=f(2x+1)=(2x+1)^{2}$
2. Suppose $h(x)={\sqrt {x}}$ and $k(x)=x^{3}$ . Then, $(h\circ k)(x)=h(k(x))=h(x^{3})={\sqrt {x^{3}}}$

Graphs of Composed Functions

To visualize the composition of functions, we can combine the graphs of the individual functions:

1. Plot the graph of $f(x)$.
2. Plot the graph of $g(x)$.
3. To find the graph of $(f\circ g)(x)$ , take the output from the graph of $g(x)$ and use it as the input for the graph of $f(x)$.

Inverse Functions

An inverse function of a function $f$ , denoted by $f^{-1}$ , is a function that reverses the operation of $f$. In other words, for any input $x$ in the domain of $f$ , there exists an output $y$ in the range of $f$ such that $f^{-1}(y)=x$.

Properties of Inverse Functions

1. Domain and Range: The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.
2. Composition: $f\circ f^{-1}=i=f^{-1}\circ f$ , where $i$ represents the identity function $(i(x)=x)$

Finding Inverse Functions

1. Replace $y$ with $x$ in the original function $f(x)$ and solve for $x$. The resulting equation represents $f^{-1}(x)$.
2. For example, if $f(x)=2x+1$ , then $f^{-1}(x)={\frac {(x-1)}{2}}$.

Applications of Composition and Inverse Functions

1. Solving equations involving composite functions
2. Modeling real-world relationships
3. Analyzing mathematical models
4. Constructing new functions with desired properties

Conclusion

Composition of functions and inverse functions are powerful tools in mathematics that allow for manipulating and transforming functions to solve problems, analyze relationships, and construct mathematical models. Understanding these concepts is essential for students to develop a deeper understanding of functions and their applications in various fields.