19
INVERSE FUNCTIONS
Definition of inverses
Constructing the inverse
Notation
The graph of an inverse function
THE INVERSE of a function undoes the action of that function.
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Say, for example, that a function f acts on 5, producing f(5). Then if g is the inverse of f, then g acting on f(5) will bring back 5 !
g(f(5)) = 5
Actually, g must do that for all values in the domain of f. And f must do that for all values in the domain of g. Here is the definition:
Functions f(x) and g(x) are inverses of one another if:
f(g(x)) = x and g(f(x)) = x,
for all values in their respective domains.
Example 1. Let f(x) = x + 2, and g(x) = x − 2. Then they are inverses of one another. For g(x), which subtracts 2 from a number, is the inverse of adding 2: f(x).
Formally, according to the definition:
f(g(x)) = f(x − 2) = (x − 2) + 2 = x,
(f adds 2 to its argument), and
g(f(x)) = g(x + 2) = (x + 2) − 2 = x.
(g subtracts 2 from its argument.)
The definition is satisfied.
Problem 1. Let f(x) = x² and g(x) = x½. Show that they are inverses of one another. (The domain of f must be restricted to x 0.)
To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload").
f(g(x)) = f(x½) = (x½)² = x,
and
g(f(x)) = g(x²) = (x²)½ = x.
Constructing the inverse
When we have a function y = f(x) -- for example
y = x²
-- then we can often "invert" the equation by solving for x. In this case,
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x now appears as a function of y. Therefore on exchanging the variables,
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is the inverse function of y = x². |
(Taking the square root of a number is the inverse of squaring a number.)
Hence, to construct the inverse of a function y = f(x):
Solve for x, then exchange the variables.
Example 2. What function is the inverse of y = 3x + 4?
Solution. Exchange the sides of the equation, and solve for x:
3x + 4 |
= |
y |
|
3x |
= |
y − 4 |
|
|
x |
= |
y − 4 3 |
| . |
|
Exchange the variables: |
|
y |
= |
x − 4 3 |
| . |
That function is the inverse of y = 3x + 4.
Problem 2. What function is the inverse of y = 5x?
On solving for x:
|
x |
= |
y 5 |
. |
|
Therefore on exchanging the variables: |
|
|
y |
= |
x 5 |
. |
Clearly, dividing by 5 is the inverse of multiplying by 5.
Problem 3. a) Let y = f(x) = x − 4. Construct its inverse, g(x).
|
x − 4 |
= |
y |
|
|
implies: |
|
|
|
x |
= |
y + 4. |
|
|
|
g(x) |
= |
x + 4. |
| |
b) Prove that f(x) and g(x) are inverses.
f(g(x)) = f(x + 4) = (x + 4) − 4 = x,
and
g(f(x)) = g(x − 4) = (x − 4) + 4 = x.
Notation
The function I(x) = x is called the identity function. It always returns x.
As a notation for the inverse of a function f, we sometimes see f −1 ("f inverse"). "−1" is not an exponent. That notation is used because in the language of composition of functions, we can write:
f o
f −1 = I
This is similar in form to the multiplication of numbers, a· a−1 = 1.
For the inverse trigonometric functions, see Topic 20 of Trigonometry.
The graph of an inverse function
The graph of the inverse of a function f(x) can be found as follows:
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Reflect the graph about the x-axis, then rotate it 90° counterclockwise.
To see that in fact that is the graph of the inverse, consider the following:
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Let A be a point on the graph of f(x) whose coordinates are (x, y), and which is a distance d from the origin C; AC makes an angle θ with the x-axis; triangle ABC is right angled.
The figure on the left shows the reflection of A about the x-axis to the point D. The figure on the right shows the rotation 90° counterclockwise to the point C'. In that figure, C'A' makes an angle of 90° − θ with the x-axis. That is, in the right triangle A'B'C', angle C'A'B' is the complement of angle θ. Therefore the angle at C' is equal to θ.
But the angle at A is the complement of θ. Therefore the triangles ABC, A'B'C' are congruent (Angle-side-angle), and those sides are equal that are opposite the equal angles:
A'B' is equal to AB -- which is the y-coordinate .
B'C' is equal to BC -- which is the x-coordinate .
Therefore the coordinates of C' are (y, x).
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In other words, when each point (x, y) on f(x) is transformed into (y, x), then a graph of a functions g(x) results. And we claim that the graph of g(x) is the graph of the inverse of f(x).
For, when g operates on its x-coordinate -- which is the y-coordinate of f -- it produces x:
g( y ) = x.
But y is f(x):
g( f(x)) = x.
And that is the definition of the inverse! The graph of g(x) is the graph of the inverse of f(x).
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What is more, we can now see that the graphs of a function and its inverse are symmetrical with respect to the straight line y = x.
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