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Describing Inverses of Functions

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

Inverse of a Function

The inverse of a function reverses its xx-yy coordinates. If, for a function f,f, xx is an input and yy is its corresponding output, for the inverse, f-1,yf^{\text{-} 1}, y is the input and xx would be the corresponding output. f(x)=yf-1(y)=x f(x)=y \quad \Leftrightarrow \quad f^{\text{-} 1}(y)=x

Some function families are inverses of each other. This is because some functions undo each other. For example, x2x^2 and ±x\pm\sqrt{x} are inverses because radicals and exponents (with the same index) undo each other.
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 f(x)=2x13. f(x)= \dfrac{2x-1}{3}.

1

Replace f(x)f(x) with yy

To begin, since f(x)=yf(x)=y describes the input-output relationship of the function, replace f(x)f(x) with yy in the function rule. f(x)=2x13y=2x13 f(x)= \dfrac{2x-1}{3} \quad \Leftrightarrow \quad y= \dfrac{2x-1}{3}

2

Switch xx and yy

Because the inverse of a function reverses xx and y,y, the variables can be switched. Notice that every other piece in the function rule remains the same. y=2x13x=2y13 {\color{#009600}{y}}= \dfrac{2 {\color{#0000FF}{x}}-1}{3} \quad \Rightarrow {\color{#0000FF}{x}}= \dfrac{2 {\color{#009600}{y}}-1}{3}

3

Solve for yy
Solve the resulting equation from Step 2 for y.y. Here, this will involve using the inverse operations.
x=2y13x=\dfrac{2y-1}{3}
3x=2y13x=2y-1
3x+1=2y3x+1=2y
3x+12=y\dfrac{3x+1}{2}=y
y=3x+12y=\dfrac{3x+1}{2}

4

Replace yy with f-1f^{\text{-} 1}

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

y=3x+12f-1=3x+12 y=\dfrac{3x+1}{2} \quad \Leftrightarrow \quad f^{\text{-} 1}=\dfrac{3x+1}{2}

Notice that in f,f, the input is multiplied by 2,2, decreased by 11 and divided by 3.3. From the rule of f-1,f^{\text{-} 1}, it can be seen that xx undergoes the inverse of these operation in the reverse order. Specifically, xx is multiplied by 3,3, increased by 1,1, and divided by 2.2.

Exercise

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

xx g(x)g(x)
-4\text{-} 4 33
-2\text{-} 2 22
00 11
22 00
44 -1\text{-} 1
Solution

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

xx g-1(x)g^{\text{-}1}(x)
33 -4\text{-}4
22 -2\text{-}2
11 00
00 22
-1\text{-}1 44

We can graph both gg and g-1g^{\text{-} 1} by marking the points from both tables on the same coordinate plane.

info Show solution Show solution
Concept

Graphs of Functions and Inverses

The graphs of a function and its inverse have a noteworthy relationship. In the coordinate plane, the function f(x)=2x3 and its inverse f-1(x)=x+32 f(x)=2x-3 \text{ and its inverse } f^{\text{-}1}(x)=\frac{x+3}{2} are graphed. Notice that the points of f-1f^{\text{-} 1} are the reversed points of f.f.

Because the coordinates of the points are reversed, f-1(x)f^{\text{-}1}(x) is a reflection of f(x)f(x) in the line y=x.y=x.

This is true for all functions and their inverses.
Concept

Invertible Function

A function is said to be invertible if its inverse is also a function. An example of an invertible function is f(x)=0.5x2, f(x)=0.5x-2, because its inverse, f-1(x)=2x+4,f^{\text{-} 1}(x)=2x+4, is a linear function. Consider the function g(x)=x2,g(x)=x^2, whose inverse is ±x.\pm\sqrt{x}.

Notice that the inverse fails the Vertical Line Test. Thus, it is not a function, which, in turn means that gg is not invertible. However, if the domain of gg is restricted to x0,x\geq 0, the inverse, g-1(x)=x,g^{\text{-} 1}(x)=\sqrt{x}, is a function.

Thus, some functions that are not inherently invertible can be made invertible by restricting their domains.
Method

Horizontal Line Test

The Vertical Line Test determines if the graph of a relation is a function, or if each xx-value corresponds with exactly one yy-value. For a function to be invertible, each y-valuemust correspond toexactly one x-value.\begin{aligned} \text{each } y & \text{-value} \\ \text{must cor} & \text{respond to} \\ \text{exactly o} & \text{ne } x\text{-value.} \end{aligned} This leads to the 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.

Perform test

Change function

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