How do you find the inverse of 2x 1?

Algebra Examples

Find the Inverse F(x)=2x-1

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Algebra Examples
1 Expert Answer
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Back to Where We Started
Solve Using Algebra
Fahrenheit to Celsius
Inverses of Common Functions
Example: Square and Square Root
But we can fix that!
Example: (continued)
No Inverse?
Domain and Range
Example: Square and Square Root (continued)
Not Always Solvable!
Notes on Notation
How do you find the inverse of an equation?
How do you solve an inverse step by step?
Does 2x have an inverse?

Interchange the variables.

Rewrite the equation as .

Add to both sides of the equation.

Divide each term in by and simplify.

Cancel the common factor of .

Cancel the common factor.

Replace with to show the final answer.

Verify if is the inverse of .

To verify the inverse, check if and .

Set up the composite result function.

Evaluate by substituting in the value of into .

Combine the numerators over the common denominator.

Combine the opposite terms in .

Cancel the common factor of .

Cancel the common factor.

Set up the composite result function.

Evaluate by substituting in the value of into .

Apply the distributive property.

Cancel the common factor of .

Cancel the common factor.

Cancel the common factor of .

Cancel the common factor.

Combine the opposite terms in .

Since and , then is the inverse of .

The inverse is #=(x-1)/2#

Explanation:

Our equation is

#y=2x+1#, we have #y# as a function of #x#

We need #x# as a function of #y#

#2x=y-1#

#x=(y-1)/2#

Therefore. the inverse is

#y^-1=(x-1)/2#

Verification

#[yoy^-1](x)=y((x-1)/2)=2*(x-1)/2+1=x#

The inverse of #y# is the image of #y# in the line #y=x#

graph{(y-x)(y-2x-1)(y-x/2+1/2)=0 [-6.24, 6.244, -3.12, 3.12]}

Explanation:

Write as: #y=(2x+1)/x#

Swap #x# and #y#.

#x=(2y+1)/y#

Solve for #y#.

#xy=2y+1#

#xy-2y=1#

#y(x-2)=1#

#y=1/(x-2)#

You can rewrite #y# as #f^-1(x)#, which denotes an inverse function.

#f^-1(x)=1/(x-2)#

Redeana M.

asked • 12/13/16

How do you do inverse functions for f(x)

Philip M. answered • 12/13/16

Math Made EASY!

To find the inverse, we switch the "x" and "y" variables in the equation and solve for the "new y". The new "y" is actually the inverse function, or y^(-1) or f^(-1)(x). Note: f(x) is the same as y, its just different notation. And y^(-1) is the same as f^(-1)(x).

y=2x-1 becomes x=2y-1.

Solving for the "new y", which we change it to y^(-1):

x=2y^(-1)-1

x+1=2y^(-1)

(x+1)/2=y^(-1).

And that's it! Your inverse function, f^(-1)(x)= (x+1)/2.

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An inverse function goes the other way!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram:

The Inverse Function goes the other way:

So the inverse of: 2x+3 is: (y-3)/2

The inverse is usually shown by putting a little "-1" after the function name, like this:

f-1(y)

We say "f inverse of y"

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

f-1(y) = (y-3)/2

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

Back to Where We Started

The cool thing about the inverse is that it should give us back the original value:

When the function f turns the apple into a banana,
Then the inverse function f-1 turns the banana back to the apple

Example:

Using the formulas from above, we can start with x=4:

f(4) = 2×4+3 = 11

We can then use the inverse on the 11:

f-1(11) = (11-3)/2 = 4

And we magically get 4 back again!

We can write that in one line:

f-1( f(4) ) = 4

"f inverse of f of 4 equals 4"

So applying a function f and then its inverse f-1 gives us the original value back again:

f-1( f(x) ) = x

We could also have put the functions in the other order and it still works:

f( f-1(x) ) = x

Example:

Start with:

f-1(11) = (11-3)/2 = 4

And then:

f(4) = 2×4+3 = 11

So we can say:

f( f-1(11) ) = 11

"f of f inverse of 11 equals 11"

Solve Using Algebra

We can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function:	f(x)	=	2x+3
Put "y" for "f(x)":	y	=	2x+3
Subtract 3 from both sides:	y-3	=	2x
Divide both sides by 2:	(y-3)/2	=	x
Swap sides:	x	=	(y-3)/2

Solution (put "f-1(y)" for "x") :	f-1(y)	=	(y-3)/2

This method works well for more difficult inverses.

Fahrenheit to Celsius

A useful example is converting between Fahrenheit and Celsius:

To convert Fahrenheit to Celsius:f(F) = (F - 32) × 59

The Inverse Function (Celsius back to Fahrenheit):f-1(C) = (C × 95) + 32

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

Inverses of Common Functions

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Here is a list to help you:

Inverses	Careful!
	<=>
	<=>		Don't divide by zero
1x	<=>	1y	x and y not zero
x2	<=>		x and y ≥ 0
xn	<=>	or	n not zero (different rules when n is odd, even, negative or positive)
ex	<=>	ln(y)	y > 0
ax	<=>	loga(y)	y and a > 0
sin(x)	<=>	sin-1(y)	-π/2 to +π/2
cos(x)	<=>	cos-1(y)	0 to π
tan(x)	<=>	tan-1(y)	-π/2 to +π/2

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

Careful!

Did you see the "Careful!" column above? That is because some inverses work only with certain values.

Example: Square and Square Root

When we square a negative number, and then do the inverse, this happens:

Square:(−2)2 = 4

Inverse (Square Root): √(4) = 2

But we didn't get the original value back! We got 2 instead of −2. Our fault for not being careful!

So the square function (as it stands) does not have an inverse

But we can fix that!

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

Example: (continued)

Just make sure we don't use negative numbers.

In other words, restrict it to x ≥ 0 and then we can have an inverse.

So we have this situation:

x2 does not have an inverse
but {x2 | x ≥ 0 } (which says "x squared such that x is greater than or equal to zero" using set-builder notation) does have an inverse.

No Inverse?

Let us see graphically what is going on here:

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

Just think ... if there are two or more x-values for one y-value, how do we know which one to choose when going back?

Imagine we came from x1 to a particular y value, where do we go back to? x1 or x2?

In that case we can't have an inverse.

But if we can have exactly one x for every y we can have an inverse.

It is called a "one-to-one correspondence" or Bijective, like this

A function has to be "Bijective" to have an inverse.

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

Domain and Range

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

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

As it stands the function above does not have an inverse, because some y-values will have more than one x-value.

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

... and now we can have an inverse:

Note also:

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

Let's plot them both in terms of x ... so it is now f-1(x), not f-1(y):

f(x) and f-1(x) are like mirror images
(flipped about the diagonal).

In other words:

The graph of f(x) and f-1(x) are symmetric across the line y=x

Example: Square and Square Root (continued)

First, we restrict the Domain to x ≥ 0:

{x2 | x ≥ 0 } "x squared such that x is greater than or equal to zero"
{√x | x ≥ 0 } "square root of x such that x is greater than or equal to zero"

And you can see they are "mirror images"
of each other about the diagonal y=x.

Note: when we restrict the domain to x ≤ 0 (less than or equal to 0) the inverse is then f-1(x) = −√x:

{x2 | x ≤ 0 }
{−√x | x ≥ 0 }

Which are inverses, too.

Not Always Solvable!

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

Example: f(x) = x/2 + sin(x)

We cannot work out the inverse of this, because we cannot solve for "x":

y = x/2 + sin(x)

y ... ? = x

Notes on Notation

Even though we write f-1(x), the "-1" is not an exponent (or power):

f-1(x)	...is different to...	f(x)-1
Inverse of the function 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 can find an inverse by using Algebra:
- Put "y" for "f(x)", and
- Solve for x
We may need to restrict the domain for the function to have an inverse

How do you find the inverse of an equation?

To find the inverse of a function, write the function y as a function of x i.e. y = f(x) and then solve for x as a function of y.

How do you solve an inverse step by step?

Steps for finding the inverse of a function f..

Replace f(x) by y in the equation describing the function..

Interchange x and y. In other words, replace every x by a y and vice versa..

Solve for y..

Replace y by f-1(x)..

Does 2x have an inverse?

In our example, the answer is clearly yes. If we multiply some number by 2, we can always divide the resulting number by two to recover our original number. Therefore, the function f(x) = 2x is said to be invertible.