Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

Inverses and One-to-One Functions

Let’s consider the following two functions.

Suppose we reverse the arrows. Are these inverse relations functions?

We see that the inverse of the postage function is not a function. Like all functions, each input in the postage function has exactly one output. However, the output for 2009, 2010, and 2011 is 44. Thus in the inverse of the postage function, the input 44 has three outputs, 2009, 2010, and 2011. When two or more inputs of a function have the same output, the inverse relation cannot be a function. In the cubing function, each output corresponds to exactly one input, so its inverse is also a function. The cubing function is an example of a one-to-one function.

If the inverse of a function f is also a function, it is named $f^{- 1}$ $f^{- 1}$ (read “f-inverse”).

The −1 in $f^{- 1}$ $f^{- 1}$ is not an exponent!

Do not misinterpret the −1 in $f^{- 1}$ $f^{- 1}$ as a negative exponent: $f^{- 1}$ $f^{- 1}$ does not mean the reciprocal of f and $f^{- 1} (x)$ $f^{- 1} (x)$ is not equal to $\frac{1}{f (x)}$ $\frac{1}{f (x)}$ .

Example 3

Given the function f described by $f (x) = 2 x - 3$ $f (x) = 2 x - 3$ , prove that f is one-to-one (that is, it has an inverse that is a function).

Solution

To show that f is one-to-one, we show that if $f (a) = f (b)$ $f (a) = f (b)$ , then $a = b$ $a = b$ . Assume that $f (a) = f (b)$ $f (a) = f (b)$ for a and b in the domain of f. Since $f (a) = 2 a - 3$ $f (a) = 2 a - 3$ and $f (b) = 2 b - 3$ $f (b) = 2 b - 3$ , we have

\begin{array}{rcl} 2 a - 3 & = & 2 b - 3 \\ 2 a & = & 2 b Adding 3 \\ a & = & b . Dividing by 2 \end{array}

$\begin{array}{rcl} 2 a - 3 & = & 2 b - 3 \\ 2 a & = & 2 b Adding 3 \\ a & = & b . Dividing by 2 \end{array}$

Thus, if $f (a) = f (b)$ $f (a) = f (b)$ , then $a = b$ $a = b$ . This shows that f is one-to-one.

Now Try Exercise 17.

Example 4

Given the function g described by $g (x) = x^{2}$ $g (x) = x^{2}$ , prove that g is not one-to-one.

Solution

We can prove that g is not one-to-one by finding two numbers a and b for which $a \neq b$ $a \neq b$ and $g (a) = g (b)$ $g (a) = g (b)$ . Two such numbers are −3 and 3, because $- 3 \neq 3$ $- 3 \neq 3$ and $g (- 3) = g (3) = 9$ $g (- 3) = g (3) = 9$ . Thus g is not one-to-one.

Now Try Exercise 21.

The following graphs show a function, in blue, and its inverse, in red. To determine whether the inverse is a function, we can apply the vertical-line test to its graph. By reflecting each such vertical line across the line $y = x$ $y = x$ , we obtain an equivalent horizontal-line test for the original function.

Example 5

From the graphs shown, determine whether each function is one-to-one and thus has an inverse that is a function.

Solution

For each function, we apply the horizontal-line test.

RESULT	REASON
One-to-one; inverse is a function Not one-to-one; inverse is not a function One-to-one; inverse is a function Not one-to-one; inverse is not a function	No horizontal line intersects the graph more than once. There are many horizontal lines that intersect the graph more than once. Note that where the line $y = 4$ $y = 4$ intersects the graph, the first coordinates are −2 and 2. Although these are different inputs, they have the same output, 4. No horizontal line intersects the graph more than once. There are many horizontal lines that intersect the graph more than once.

RESULT

REASON

One-to-one; inverse is a function
Not one-to-one; inverse is not a function
One-to-one; inverse is a function
Not one-to-one; inverse is not a function

No horizontal line intersects the graph more than once.

There are many horizontal lines that intersect the graph more than once. Note that where the line $y = 4$ $y = 4$ intersects the graph, the first coordinates are −2 and 2. Although these are different inputs, they have the same output, 4.

No horizontal line intersects the graph more than once.

There are many horizontal lines that intersect the graph more than once.

Now Try Exercises 25 and 27.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Inverses and One-to-One Functions

Create new playlist

Sign In

Sign Up

Table of Contents for
Inverses and One-to-One Functions