## Introduction to Inverse Functions

To find the inverse function, switch the

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### Learning Objectives

Calculate the formula of an function’s inverse by switching

### Key Takeaways

Key PointsAn inverse function reverses the inputs and outputs.To find the inverse formula of a function, write it in the form of**inverse function**: A function that does exactly the opposite of another

### Definition of Inverse Function

An inverse function, which is notated

Below is a mapping of function

**Inverse functions:** mapping representation: An inverse function reverses the inputs and outputs.

Thus the graph of

**Inverse functions:** graphic representation: The function graph (red) and its inverse function graph (blue) are reflections of each other about the line

### Write the Inverse Function

In general, given a function, how do you find its inverse function? Remember that an inverse function reverses the inputs and outputs. So to find the inverse function, switch the

### Example 1

Find the inverse of:

a.: Write the function as:

b.: Switch the

c.: Solve for

Since the function

**The inverse is not a function:** A function’s inverse may not always be a function. The function (blue)

### Example 2

Find the inverse function of:

As soon as the problem includes an exponential function, we know that the logarithm reverses exponentiation. The complex logarithm is the inverse function of the exponential function applied to complex numbers. Let’s see what happens when we switch the input and output values and solve for

a.: Write the function as:

b.: Switch the

c.: Solve for

**Exponential and logarithm functions:** The graphs of

Test to make sure this solution fills the definition of an inverse function.

Pick a number, and plug it into the original function.## Composition of Functions and Decomposing a Function

Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.

### Learning Objectives

Practice functional composition by applying the rules of one function to the results of another function

### Key Takeaways

Key PointsFunctional composition applies one function to the results of another.Functional decomposition resolves a functional relationship into its constituent parts so that the original function can be reconstructed from those parts by functional composition.Decomposition of a function into non-interacting components generally permits more economical representations of the function.The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:**codomain**: The target space into which a function maps elements of its domain. It always contains the range of the function, but can be larger than the range if the function is not subjective.

**domain**: The set of all points over which a function is defined.

### Function Composition

The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a *composite function*. We represent this combination by the following notation:

We read the left-hand side as “*composition operator*. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number.

### Function Composition and Evaluation

It is important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside.

In general,

Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

### Evaluating Composite Functions Using Input Values

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

### Example 1

If

To evaluate

Therefore,

To evaluate

Therefore,

### Evaluating Composite Functions Using a Formula

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition

In the next example we are given a formula for two composite functions and asked to evaluate the function. Evaluate the inside function using the input value or variable provided. Use the resulting output as the input to the outside function.

### Example 2

If

First substitute, or input the function

For

### Functional Decomposition

Functional decomposition broadly refers to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition. In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components (which may reflect individual physical processes of interest), or for the purpose of obtaining a compressed representation of the global function; a task which is feasible only when the constituent processes possess a certain level of modularity (*i.e.*, independence or non-interaction).

In general, functional decompositions are worthwhile when there is a certain “sparseness” in the dependency structure; *i.e*. when constituent functions are found to depend on approximately disjointed sets of variables. Also, decomposition of a function into non-interacting components generally permits more economical representations of the function.

## Restricting Domains to Find Inverses

Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.

### Key Takeaways

Key Points**domain**: The set of points over which a function is defined.

### Inverse Functions

**Inverse functions’ domain and range:** If

### Domain Restrictions: Parabola

Informally, a restriction of a function is the result of trimming its domain. Remember that:

If

Without any domain restriction,

**Failure of horizontal line test:** Graph of a parabola with the equation

### Domain Restriction: Exponential and Logarithmic Functions

Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse. The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

### Example 1

Is

No, the function has no defined value for

## Inverses of Composite Functions

A composite function represents, in one function, the results of an entire chain of dependent functions.

### Key Takeaways

Key PointsThe composition of functions is always associative. That is, if**composite function**: A function of one or more independent variables, at least one of which is itself a function of one or more other independent variables; a function of a function

### Composition and Composite Functions

In mathematics, *function composition* is the application of one function to the results of another.

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**Composition of functions:**