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The inverse of a function can itself be a function. In this section, properties of inverses and *invertible functions* will be explored.

Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function f is another function $f_{-1}$ such that they *undo* each other.

$f(f_{-1}(x))=xandf_{-1}(f(x))=x$

Also, if x is the input of a function f and y its corresponding output, then y is the input of $f_{-1}$ and x its corresponding output.

$f(x)=2x−3andf_{-1}(x)=2x+3 $

These functions will be shown to $2f_{-1}(x)−3=?x$

Substitute

$f_{-1}(x)=2x+3 $

$2(2x+3 )−3=?x$

Simplify left-hand side

DenomMultFracToNumber

$2⋅2a =a$

$(x+3)−3=?x$

RemovePar

Remove parentheses

$x+3−3=?x$

SubTerm

Subtract term

$x=x✓$

Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|

$f(f_{-1}(x))=?x$ | $2f_{-1}(x)−3=?x$ | $2(2x+3 )−3=?x$ | $x=x✓$ |

$f_{-1}(f(x))=?x$ | $2f(x)+3 =?x$ | $22x−3+3 =?x$ | $x=x✓$ |

Therefore, f and $f_{-1}$ *undo* each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of $f_{-1}$ are the *reversed* points on the graph of f.

Depending on how a function is presented, finding its inverse can be done in different ways. When a function's rule is given, finding the inverse algebraically is advantageous. Consider the function
### 1

To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule.
### 2

Because the inverse of a function reverses x and y, the variables can be switched. Notice that every other piece in the function rule remains the same.
### 3

Solve the resulting equation from Step 2 for y. Here, this will involve using the inverse operations.
### 4

Replace f(x) with y

Switch x and y

Solve for y

$x=32y−1 $

MultEqn

LHS⋅3=RHS⋅3

3x=2y−1

AddEqn

LHS+1=RHS+1

3x+1=2y

DivEqn

$LHS/2=RHS/2$

$23x+1 =y$

RearrangeEqn

Rearrange equation

$y=23x+1 $

Replace y with $f_{-1}(x)$

Just as f(x)=y shows the input-output relationship of f, so does $f_{-1}(x)=y.$ Thus, replacing y with $f_{-1}(x)$ gives the rule for the inverse of f.

Notice that in f, the input is multiplied by 2, decreased by 1 and divided by 3. From the rule of $f_{-1},$ it can be seen that x undergoes the inverse of these operation in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.

Some of the coordinates of the function g are shown in the table. Find $g_{-1},$ then graph g and $g_{-1}$ on the same coordinate plane.

x | g(x) |
---|---|

-4 | 3 |

-2 | 2 |

0 | 1 |

2 | 0 |

4 | -1 |

Show Solution

An inverse of a function reverses its x- and y-coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point (-4,3) on g becomes (3,-4) on $g_{-1}.$ The following table describes $g_{-1}(x).$

x | $g_{-1}(x)$ |
---|---|

3 | -4 |

2 | -2 |

1 | 0 |

0 | 2 |

-1 | 4 |

We can graph both g and $g_{-1}$ by marking the points from both tables on the same coordinate plane.

The graphs of a function and its inverse have a noteworthy relationship. In the coordinate plane, the function

$f(x)=2x−3and its inversef_{-1}(x)=2x+3 $

are graphed. Notice that the points of $f_{-1}$ are the reversed points of f.
Because the coordinates of the points are reversed, $f_{-1}(x)$ is a reflection of f(x) in the line y=x.

This is true for all functions and their inverses.
A function is said to be invertible if its inverse is also a function. An example of an invertible function is

f(x)=0.5x−2,

because its inverse, $f_{-1}(x)=2x+4,$ is a linear function.
Consider the function g(x)=x2, whose inverse is $±x .$ Notice that the inverse fails the Vertical Line Test. Thus, it is not a function, which, in turn means that g is not invertible. However, if the domain of g is restricted to x≥0, the inverse, $g_{-1}(x)=x ,$ is a function.

Thus, some functions that are not inherently invertible can be made invertible by restricting their domains.
The Vertical Line Test determines if the graph of a relation is a function, or if each x-value corresponds with exactly one y-value. For a function to be invertible,
**Horizontal** Line Test, which is performed by moving an imaginary horizontal line across the graph of a function. If this line intersects the graph more than once anywhere, the function is not invertible.

$eachymust corexactly o -valuerespond tonex-value. $

This leads to the Perform test

Change function

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