Algebra ExamplesFind the Inverse F(x)=2x1 Show
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 #=(x1)/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=y1# #x=(y1)/2# Therefore. the inverse is #y^1=(x1)/2# Verification #[yoy^1](x)=y((x1)/2)=2*(x1)/2+1=x# The inverse of #y# is the image of #y# in the line #y=x# graph{(yx)(y2x1)(yx/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# #xy2y=1# #y(x2)=1# #y=1/(x2)# You can rewrite #y# as #f^1(x)#, which denotes an inverse function. #f^1(x)=1/(x2)#
Redeana M. asked • 12/13/16How do you do inverse functions for f(x) More 1 Expert Answer
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=2x1 becomes x=2y1. 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. Still looking for help? Get the right answer, fast.ORFind an Online Tutor Now Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. 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: (y3)/2 The inverse is usually shown by putting a little "1" after the function name, like this: f1(y) We say "f inverse of y" So, the inverse of f(x) = 2x+3 is written: f1(y) = (y3)/2 (I also used y instead of x to show that we are using a different value.) Back to Where We StartedThe cool thing about the inverse is that it should give us back the original value: 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: f1(11) = (113)/2 = 4 And we magically get 4 back again! We can write that in one line: f1( f(4) ) = 4 "f inverse of f of 4 equals 4" So applying a function f and then its inverse f1 gives us the original value back again: f1( f(x) ) = x We could also have put the functions in the other order and it still works: f( f1(x) ) = x Example:Start with: f1(11) = (113)/2 = 4 And then: f(4) = 2×4+3 = 11 So we can say: f( f1(11) ) = 11 "f of f inverse of 11 equals 11" Solve Using AlgebraWe can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:
This method works well for more difficult inverses. Fahrenheit to CelsiusA 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):f1(C) = (C × 95) + 32 For you: see if you can do the steps to create that inverse! Inverses of Common FunctionsIt 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:
(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 RootWhen 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:
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 xvalues for one yvalue, 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 "onetoone 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 RangeSo 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 yvalues will have more than one xvalue.
But we could restrict the domain so there is a unique x for every y ... ... and now we can have an inverse: Note also:
Let's plot them both in terms of x ... so it is now f1(x), not f1(y): f(x) and f1(x) are like mirror images In other words: The graph of f(x) and f1(x) are symmetric across the line y=x Example: Square and Square Root (continued)First, we restrict the Domain to x ≥ 0:
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 f1(x) = −√x:
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 NotationEven though we write f1(x), the "1" is not an exponent (or power):
Summary
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 f1(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.
