Inverse Functions Help

Introduction to Inverse Functions

In the same way operations on real numbers (like addition and multiplication) have identities and inverses, operations on functions can have identities and inverses. We can apply many operations on functions that we can apply to real numbers—adding, multiplying, raising to powers, etc. These operations can have identities and functions have inverses in the same way they do with real numbers. The additive identity for function addition is i(x) = 0. Each function has an additive inverse, − f(x) is the additive inverse for f (x). The multiplicative identity for function multiplication is i(x) = 1, and the multiplicative inverse for Combinations of
Functions and Inverse Functions Solutions .

If we look at function composition as an operation on functions, then we can ask whether or not there is an identity for this operation and whether or not functions have inverses for this operation. There is an identity for this operation, i (x) = x. For any function f (x), f o i (x) = f ( i (x)) = f (x). Some functions have inverses. Later we will see which functions have inverses and how to find inverses. The notation for the inverse function of f (x) is f⁻¹ (x). This is different from (f (x)) ⁻¹ , which is the multiplicative inverse for f (x). For now, we will be given two functions that are said to be inverses of each other. We will use function composition to verify that they are.

Examples

Verify that f (x) and g (x) are inverses.

f (x) = 2 x + 5 and

We will show that f o g (x ) = x and g o f(x) = x.

If we think of a function as a collection of points on a graph, or ordered pairs, then the only thing that makes f(x) different from f ⁻¹(x) is that their x -coordinates and y -coordinates are reversed. For example, if (3, − 1) is a point on the graph of f(x), then (−1, 3) is a point on the graph of f ⁻¹( x ).

Example

The graph of a function f ( x ) is given in Figure 4.3. Use the graph of f ( x ) to sketch the graph of f ⁻¹ ( x ).

Combinations of Functions and Inverse
Functions Example

Fig. 4.3

We will make a table of values for f ( x ) and switch the x and y columns for f ⁻¹ ( x ).

Table 4.1

x	y = f(x)
−5	−3
−3	0
0	1
1	3
5	5

To get the table for f ⁻¹ ( x ), we will switch the x - and y -values.

Table 4.2

x	y = f ⁻¹ (x)
−3	−5
0	−3
1	0
3	1
5	5

The solid graph is the graph of f ( x ), and the dashed graph is the graph of f ⁻¹ ( x ).

Combinations of Functions and Inverse
Functions Example

Fig. 4.4

If f ( x ) is a function that has an inverse, then the graph of f ⁻¹ ( x ) is a reflection of the graph of f ( x ) across the line y = x .

Combinations of Functions and Inverse
Functions Example

Fig. 4.5

Combinations of Functions and Inverse
Functions Example

Fig. 4.6

One-To-One Functions

A function has an inverse if its graph passes the Horizontal Line Test—if any horizontal line touches the graph in more than one place, then the function will not have an inverse. Functions whose graphs pass the Horizontal Line Test are called one-to-one functions. For a one-to-one function, every x will be paired with exactly one y and every y will be paired with exactly one x .

Example

The graph of f ( x ) is given in Figure 4.7. Is f ( x ) one to one?

Fig. 4.7

This graph fails the Horizontal Line Test, so f ( x ) is not one to one.

Fig. 4.8

For functions that are not one to one, we can restrict the domain to force the function to be one to one. The function whose graph is in Figure 4.9, f ( x ) = x ² −3, is not one to one. If we restrict the domain to x ≥ 0, then the new function is one to one.

Finding the inverse function is not hard, but it can be a little tedious. The steps below show the process of algebraically switching x and y.