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Miscellaneous Functions and Equations

Finding Inverses of Functions

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 such that they undo each other.

Also, if x is the input of a function f and y its corresponding output, then y is the input of and x its corresponding output.

Example

Consider a function f and its inverse
These functions will be shown to undo each other. To do so, it needs to be proven that and that To start, the first equality will be proven. First, the definition of f will be used.
Now, in the above equation, will be substituted for
Simplify left-hand side
A similar procedure can be performed to show that
Definition of First Function Substitute Second Function Simplify

Therefore, f and 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 are the reversed points on the graph of f.

inverse function
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Exercise

Some of the coordinates of the function g are shown in the table. Find then graph g and on the same coordinate plane.

x g(x)
-4 3
-2 2
0 1
2 0
4 -1
Show Solution
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 The following table describes

x
3 -4
2 -2
1 0
0 2
-1 4

We can graph both g and by marking the points from both tables on the same coordinate plane.

Method

Finding the Inverse of a Function

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

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

2

Switch x and y
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 for y
Solve the resulting equation from Step 2 for y. Here, this will involve using the inverse operations.
3x=2y1
3x+1=2y

4

Replace y with

Just as f(x)=y shows the input-output relationship of f, so does Thus, replacing y with 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 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.

fullscreen
Exercise

Consider the quadratic function f(x)=3x2. Find its inverse function for when x>0.

Show Solution
Solution
To find the inverse of f(x), we first need to replace f(x) with y.
The next step is to switch x and y in the function rule.
Now we need to solve for y. The resulting equation will be the inverse of the given function.
x=3y2
Switching x and y also switches the domain and the range. This means that the restriction on the domain of the function, x>0, becomes a restriction on the range of the inverse function, y>0. Thus, there are no negative y-values.
Finally, to indicate that this is the inverse of f(x), we will replace y with
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