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P R E C A L C U L U S

INVERSE FUNCTIONS

Definition of inverses

Extracting the argument

Constructing the inverse

Notation

The graph of an inverse function

THE INVERSE of a function reverses the action of that function.

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 of x in their respective domains.

A definition states the conditions for the use of a word or a name. Here, if

f(g(x)) = x and g(f(x)) = x,

then we may say that the functions f and g are "inverses."

Conversely, if we say that f and g are inverses, then we would know that

f(g(x)) = x and g(f(x)) = x,

A definition is implicitly an if and only if sentence.

See First Principles of Euclid's Elements, Commentary on the Definitions.

Problem 1. Let f(x) and g(x) be inverses. Then if

f(0) = 8,

what is the value of g(8)?

Example 1. Addition and subtraction are inverses. Subtracting a specific number reverses, or undoes, the result of adding it.

In the language of functions, let

f(x) = x + 2, and g(x) = x − 2.

f(x) adds 2 to its argument. g(x) subtracts 2.

Upon applying the definition:

f(g(x)) = f(x − 2) = (x − 2) + 2 = x,

and

g(f(x)) = g(x + 2) = (x + 2) − 2 = x.

The definition is satisfied. The functions f and g are inverses.

Problem 2. Let f(x) = x² and g(x) = x^½. Show that they are inverses of one another. (The domain of f must be restricted to x 0.)

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Extracting the argument

When we write

(x + 3)⁴,

then x + 3 is the argument of the function

f(x) = x⁴.

f is that function which takes the 4th power of its argument.

Its inverse, g(x), will take the 4th root.

g(x) = x^¼.

Example 2. Solve for x:

(x + 3)⁴ = 16.

Solution. To do that, we must free, or extract, the argument x + 3. We must write

x + 3 = . . .

And to extract the argument of any function, we simply take its inverse. In this example, we take the 4th root of both sides of the equation. We can immediately write

x + 3 = 16^¼ = 2.

Therefore,

x = 2 − 3 = −1.

Problem 3. Solve for 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,

x now appears as a function of y. Therefore on exchanging the variables,

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 3. 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 4. What function is the inverse of y = 5x?

	x	=	y 5	.

On exchanging the variables:

	y	=	x 5	.

Clearly, dividing by 5 is the inverse of multiplying by 5.

Problem 5. 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.

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 19 of Trigonometry.

The graph of an inverse function

The graph of the inverse of a function f(x) can be found as follows:

Reflect the graph about the x-axis, then rotate it 90° counterclockwise

(If we take the graph of the left to be the right-hand branch of y = x², then the graph on the right is its inverse, y = .)

To see that in fact that is the graph of the inverse, let A be any point on

the graph of f(x), let its coördinates be (a, b), let it be a distance d from the origin C, and let AC make 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 of D 90° counterclockwise to the point C'.

We will see that the coördinates of C' are (b, a) -- and those are coördinates on the graph of the inverse of f (x) For if we call that inverse g(x), then according to the figure on the left,

f (a) = b.

And g(b) -- the figure on the right -- returns us to a:

g(b) = a.

The definition of the inverse is satisfied.

To see that the coördinates of C' are (b, a), consider that since angle C'A'D is 90°, then C'A' makes an angle of 90° − θ with the x-axis. That is, angle C'A'B' is the complement of angle B'A'D, which is angle θ. Therefore in the right triangle A'B'C', 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 b, the y-coördinate of f (x).

B'C' is equal to BC -- which is a, the x-coördinate of f (x).

Therefore the coördinates of C' are (b, a).

So, when each point (a, b) on f(x) is transformed into (b, a), then the graph that results is its inverse.

Each point (a, b) will also be transformed into (b, a) when (a, b) is reflected about the line y = x.

Therefore we say that the graphs of a function and its inverse are symmetrical with respect to the straight line y = x.

Next Topic: Logarithms

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