What is the Value of F 1

September 8, 2022 admin 114 Views

Inverse Functions

An inverse function goes the other fashion!

Let united states beginning with an case:

Here nosotros have the function
f(x) = 2x+3, written every bit a flow diagram:

The
Changed Office
goes the other manner:

Inverse

Then the changed of: 2x+3 is: (y-3)/2

The changed is ordinarily shown by putting a piffling “-1” after the function proper noun, similar this:

f^-1(y)

We say “f inverse
of y”

So, the inverse of
f(ten) = 2x+3
is written:

f^-i(y) = (y-3)/2

(I also used
y
instead of
ten
to show that we are using a different value.)

Dorsum to Where We Started

The cool matter about the inverse is that information technology should requite us back the original value:

banana f() to apple f()-inverse back to banana

When the function
f
turns the apple into a banana,

And then the
inverse
function
f^-1

turns the banana back to the apple

Example:

Using the formulas from to a higher place, we tin can showtime with
x=4:

f(4) = two×4+three = 11

We tin can and so use the changed on the eleven:

f^-1(11) = (eleven-3)/ii = 4

And we magically become
4
back again!

We can write that in ane line:

f^-1( f(4) ) = iv

“f inverse of f of 4 equals 4”

Then applying a function
f
and then its inverse
f^-1

gives the states the original value dorsum again:

f^-1( f(x) ) = x

We could also have put the functions in the other guild and information technology still works:

f( f^-one(ten) ) = 10

Instance:

Start with:

f^-i(11) = (11-three)/ii = 4

So:

f(4) = 2×4+3 = 11

So nosotros can say:

f( f^-one(11) ) = 11

“f of f inverse of xi equals 11”

Solve Using Algebra

We can work out the inverse using Algebra.
Put “y” for “f(10)” and solve for ten:

The function:	f(ten)	=	2x+3
Put “y” for “f(x)”:	y	=	2x+3
Subtract three from both sides:	y-iii	=	2x
Divide both sides past ii:	(y-3)/2	=	10
Swap sides:	10	=	(y-3)/2

Solution (put “f^-1(y)” for “x”) :	f^-ane(y)	=	(y-three)/two

This method works well for more hard inverses.

Fahrenheit to Celsius

A useful example is converting between Fahrenheit and Celsius:

To convert Fahrenheit to Celsius:

f(F) = (F – 32) ×

five
ix

The
Changed Function
(Celsius dorsum to Fahrenheit):

f^-1(C) = (C ×

nine
5
) + 32

For you: see if you can do the steps to create that inverse!

Inverses of Common Functions

Information technology has been easy and then far, because we know the inverse of Multiply is Divide, and the inverse of Add is Decrease, merely what about other functions?

Hither is a list to help you:

Inverses			Careful!
	<=>
	<=>		Don’t divide by zero
1 10	<=>	1 y	x and y not zero
ten²	<=>		x and y ≥ 0
xⁿ	<=>	or	northward not zero (different rules when northward is odd, even, negative or positive)
e^ten	<=>	ln(y)	y > 0
a^x	<=>	log_a(y)	y and a > 0
sin(x)	<=>	sin^-1(y)	–π/2 to +π/two
cos(x)	<=>	cos^-1(y)	0 to π
tan(x)	<=>	tan^-1(y)	–π/2 to +π/2

(Notation: you lot tin read more than near Inverse Sine, Cosine and Tangent.)

Careful!

Did you lot see the “Conscientious!” cavalcade above? That is because some inverses work
just with certain values.

Example: Square and Square Root

When we square a
negative
number, and so practise the inverse, this happens:

Square:
(−2)²
= iv

Inverse (Foursquare Root):

√(4) =
2

But we didn’t become the original value back! We got
2
instead of
−2. Our fault for non existence careful!

And then the square part (every bit it stands)
does not have an inverse

Just we can set up that!

Restrict the Domain
(the values that can go into a function).

Example: (connected)

Just brand sure nosotros don’t use negative numbers.

In other words, restrict information technology to
x ≥ 0
and so we can have an inverse.

So we have this situation:

x²

does
non
have an inverse
but
{x²
| x ≥ 0 }
(which says “10 squared such that x is greater than or equal to cipher” using set-architect note)
does
have an inverse.

No Inverse?

Let usa run into graphically what is going on hither:

To be able to have an inverse we demand
unique values.

Just think … if in that location are 2 or more
x-values
for one
y-value, how practise nosotros know which one to choose when going back?

Imagine we came
from
x₁
to a particular y value, where do nosotros go back to? x₁
or 10₂?

In that case nosotros can’t have an changed.

Simply if we can take exactly one ten for every y nosotros can accept an inverse.

It is chosen a “one-to-one correspondence” or Bijective, like this

A office has to be “Bijective” to have an inverse.

So a bijective function follows stricter rules than a general function, which allows us to take an inverse.

Domain and Range

So what is all this talk about “Restricting the Domain“?

domain and range graph

In its simplest form the
domain
is all the values that go into a office (and the
range
is all the values that come out).

Every bit information technology stands the part above does
not
accept an changed, because some y-values volition have more than one 10-value.

doman and range

But we could restrict the domain so there is a
unique x for every y
…

doman and range

… and now we
can
have an inverse:

Note also:

The function
f(x)
goes from the domain to the range,
The inverse part
f^-1(y)
goes from the range back to the domain.

Permit’s plot them both in terms of
x
… and so information technology is now
f^-i(10), not
f^-1(y):

doman and range

f(x)
and
f^-1(x)
are like mirror images

(flipped almost the diagonal).

In other words:

The graph of
f(10)
and
f^-i(10)
are symmetric beyond the line
y=x

Instance: Square and Square Root (continued)

First, we restrict the Domain to
ten ≥ 0:

{x^two
| x ≥ 0 }
“x squared such that x is greater than or equal to zero”
{√x | x ≥ 0 }
“square root of x such that ten is greater than or equal to goose egg”

x^2 vs square root x

And you tin can see they are “mirror images”

of each other about the diagonal y=x.

x^2 vs -square root x

Note: when we restrict the domain to
x ≤ 0
(less than or equal to 0) the inverse is then
f^-i(ten) = −√x:

{10^two
| x ≤ 0 }
{−√10 | 10 ≥ 0 }

Which are inverses, too.

Not E’er Solvable!

Information technology is sometimes not possible to find an Inverse of a Function.

Instance:
f(ten) = x/2 + sin(x)

We cannot work out the inverse of this, because we cannot solve for “10”:

y = ten/2 + sin(x)

y … ? = x

Notes on Notation

Even though nosotros write
f^-one(x), the “-one” is
not
an exponent (or power):

f^-1(x)	…is different to…	f(x) ^-ane
Changed of the part f		f(x)^-1 = 1/f(x) (the Reciprocal)

Summary

The inverse of f(x) is f^-1(y)

We can find an inverse by reversing the “flow diagram”

Or we tin can find an inverse by using Algebra:

Put “y” for “f(10)”, and
Solve for x

We may need to
restrict the domain
for the part to have an inverse