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### Inverse Functions and Relations

- If you know the graph of f, use the horizontal line test on that graph to determine whether f has an inverse.
- If you know the graphs of f and g, these functions are inverses of each other if and only if their graphs are mirror images across the line y = x.
- Two functions are inverses of each other if and only if both of their compositions are the identity function f(x) = x for all x.

### Inverse Functions and Relations

- To find the inverse of the relaion R, you must switch the Range and the Domain.

^{ − 1}= { ( − 3, − 2 ),( 2, − 1 ),( − 2,0 ),( 4,2 ),( 8,6 ),( 0,8 )}

- To find the inverse of the relaion R, you must switch the Range and the Domain.

^{ − 1}= { ( − 2, − 4 ),( 1, − 2 ),( − 3, − 1 ),( 3,1 ),( 1,4 ),( 9,6 )}

- Step 1: Change f(x) to y
- y = 3x + 2
- Step 2: Interchange x and y
- x = 3y + 2
- Step 3: Solve for y
- x = 3y + 2
- − 2 + x = 3y
- [( − 2 + x)/3] = [3y/3]
- y = [1/3]x − [2/3]
- Step 4: Change y to f
^{ − 1}(x)

^{ − 1}(x) = [1/3]x − [2/3]

- Step 1: Change f(x) to y
- y = − 5x − 3
- Step 2: Interchange x and y
- x = − 5y − 3
- Step 3: Solve for y
- x = − 5y − 3
- x + 3 = − 5y
- [(x + 3)/( − 5)] = [( − 5y)/( − 5)]
- y = − [1/5]x + [3/5]
- Step 4: Change y to f
^{ − 1}(x)

^{ − 1}(x) = − [1/5]x + [3/5]

- Step 1: Change f(x) to y
- y = [3/2]x + 5
- Step 2: Interchange x and y
- x = [3/2]y + 5
- Step 3: Solve for y
- x − 5 = [3/2]y
- Multiply both sides of the equal sign by the reciprocal of the coefficient of y.
- [2/3]( x − 5 ) = ( [3/2]y )[2/3]
- [2/3]( x − 5 ) = ( [/]y )[/]
- [2/3]( x − 5 ) = y
- y = [2/3]x − [10/3]
- Step 4: Change y to f
^{ − 1}(x)

^{ − 1}(x) = [2/3]x − [10/3]

- Step 1: Change f(x) to y
- y = − [1/9]x + 3
- Step 2: Interchange x and y
- x = − [1/9]y + 3
- Step 3: Solve for y
- x − 3 = − [1/9]y
- Multiply both sides of the equal sign by the reciprocal of the coefficient of y.
- − [9/1]( x − 3 ) = ( − [1/9]y ) − [9/1]
- [( − 9)/1]( x − 3 ) = ( [/]y )[/]
- [( − 9)/1]( x − 3 ) = y
- y = − 9x + 27
- Step 4: Change y to f
^{ − 1}(x)

^{ − 1}(x) = − 9x + 27

Are f and g inverses of each other?

- f& g are inverses of each other if and only if (f °g)(x) = (g °f)(x) = x
- Step 1: Find (f °g)(x)
- (f °g)(x) = f(g(x)) = − [1/3](g(x)) − [4/3] = − [1/3]( − 5x − 25) − [4/3] = [5/3]x + [25/3]

Are f and g inverses of each other?

- f& g are inverses of each other if and only if (f °g)(x) = (g °f)(x) = x
- Step 1: Find (f °g)(x)
- (f °g)(x) = f(g(x)) = − (g(x)) − 3 = − (1 − [2/3]x) − 3 = − 1 + [2/3]x

Are f and g inverses of each other?

- f& g are inverses of each other if and only if (f °g)(x) = (g °f)(x) = x
- Step 1: Find (f °g)(x)
- (f °g)(x) = f(g(x)) = 1 − [1/4](g(x)) = 1 − [1/4]( − 4x + 4) = 1 + x − 1 = x
- Step 2: Find (g °f)(x)
- (g °f)(x) = g(f(x)) = − 4(f(x)) + 4 = − 4(1 − [1/4]x) + 4 = − 4 + x + 4 = x

Are f and g inverses of each other?

- f& g are inverses of each other if and only if (f °g)(x) = (g °f)(x) = x
- Step 1: Find (f °g)(x)
- (f °g)(x) = f(g(x)) = − [1/3](g(x)) + [5/3] = − [1/3]( − 3x + 5) + [5/3] = x − [5/3] + [5/3] = x
- Step 2: Find (g °f)(x)
- (g °f)(x) = g(f(x)) = − 3(f(x)) + 5 = − 3( − [1/3]x + [5/3]) + 5 = x − 5 + 5 = x

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Inverse Functions and Relations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Inverse of a Relation
- Inverse of a Function
- Procedure to Construct an Inverse Function
- f(x) to y
- Interchange x and y
- Solve for y
- Write Inverse f(x) for y
- Example: Inverse Function
- Example: Inverse Function 2
- Inverses and Compositions
- Example 1: Inverse Relation
- Example 2: Inverse of Function
- Example 3: Inverse of Function
- Example 4: Inverse Functions

- Intro 0:00
- Inverse of a Relation 0:14
- Example: Ordered Pairs
- Inverse of a Function 3:24
- Domain and Range Switched
- Example: Inverse
- Procedure to Construct an Inverse Function 6:42
- f(x) to y
- Interchange x and y
- Solve for y
- Write Inverse f(x) for y
- Example: Inverse Function
- Example: Inverse Function 2
- Inverses and Compositions 10:44
- Example: Inverse Composition
- Example 1: Inverse Relation 14:49
- Example 2: Inverse of Function 15:40
- Example 3: Inverse of Function 17:06
- Example 4: Inverse Functions 18:55

### Algebra 2

### Transcription: Inverse Functions and Relations

*Welcome to Educator.com.*0000

*Today we will be covering inverse functions and relations.*0002

*And this is a topic (inverse relations) that we introduced a little bit in Algebra I.*0006

*But we will be going into much greater depth today.*0011

*First, reviewing the definition of the inverse of a relation: recall that a relation is a set of ordered pairs.*0014

*The inverse relation of relation R (and the notation is this, R ^{-1}, or if you had a different letter,*0022

*when we are talking about functions, then you would put that letter, and then -1,*0033

*to express that you are talking about the inverse of a relation) is the set of ordered pairs*0037

*in which the coordinates of each ordered pair of R are reversed.*0047

*So, let's think about what this means.*0051

*First of all, a relation is a set of ordered pairs; and that would be something like {(2,4),(3,-7),(2,6),(8,-4)}.*0056

*And this is a relation, because there is a correspondence between the first element and the second element.*0070

*You could write these out as a table; and each of the first elements has a correspondence with a member of the second set.*0078

*So, this is the relation; the inverse of R, R ^{-1}, would be when the first and second coordinates are switched.*0088

*What we are doing, then, is interchanging the domain and the range.*0101

*So, 2, 3, 2, and 8 are members of the domain; and 4, -7, 6, -4 are members of the range; now it becomes the opposite.*0105

*Now, 4, -7, 6, -4 are the domain, and these other values are the range.*0127

*I also want to point something out: let's write R as a table, and I have my x-values, 2, 3, 2, 8, and my y-values, 4, -7, 6, -4.*0134

*This is a relation, because there is a correspondence between members of the x (the domain), and members of the range (the y-values).*0149

*2 corresponds to 4, 3 to -7, and so on.*0158

*So, this is a relation; however, it is not a function; and that is because you see that 2 is assigned to two values of the range.*0162

*And in a function, each member of the domain is assigned to only one member of the range.*0175

*So, this is a relation--yes; function--no; it is not a function.*0183

*Functions are relations, but not all relations are functions; functions are a subset of relations.*0196

*Let's talk more about functions right now.*0201

*The inverse function (f ^{-1}(x)) is a special case of an inverse relation; the inverse of a function f(x) is a special case.*0204

*And, as I talked about, there are some restrictions on functions.*0216

*Functions must meet the criteria that each element of the domain is assigned to only one member of the range.*0222

*f ^{-1}(b) = a if and only if f(a) = b.*0232

*The domain and range of the inverse function are the range and domain of the function.*0240

*Just as I talked about with relations, the domain and range are flipped; they are switched around*0247

*when you go from the function to the inverse, or from the inverse to the function.*0255

*And let's think about what this is saying right here: f ^{-1}(b) = a if and only if f(a) = b.*0260

*Let's look at what that would mean in a general case.*0269

*If I have some function, and then I said part of this function is the ordered pair (3,4),*0276

*if it has an inverse, then that means that the values are going to be reversed.*0285

*If I looked for f(3), I would get 4 out; if I went ahead and put 4 into the inverse, my result would be 3; domain and range are switched around.*0296

*Taking a more specific example, if f(x) = 2x - 1, and f ^{-1}(x) is (x + 1)/2, let's say I was asked to find f(6).*0310

*Then, I would substitute that 6 in here to get 11.*0331

*My expectation would be that f ^{-1}(11) would be 6, because, if f(a) = b,*0338

*and here we are letting a equal 6 and b equal 11, then f ^{-1}(b), f^{-1}(11), should equal 6.*0349

*So, let's check out if that holds up--let's see if it does.*0362

*I am asked to find f ^{-1}(11), which would be 11 + 1, over 2, equals 12/2, equals 6.*0364

*So, f ^{-1}(11) is 6, and f(6) is 11; so, this holds up and supports the fact that these two are inverse functions of each other.*0377

*There is a certain set of rules that you can follow to actually construct an inverse function.*0402

*So, if f(x) is given, f ^{-1} can be constructed by following this procedure.*0407

*First, write y instead of f(x): change the f(x) notation to y.*0412

*Then, interchange x and y: wherever there is an x, make it a y; wherever there is a y, change that to an x.*0418

*The hardest step is this one: you need to then solve for y, which, depending on the function, can be somewhat complicated.*0427

*Then, write f ^{-1}(x) for y: once you have solved for y (you have isolated y on the left side of the equation),*0434

*you rewrite that as f ^{-1}, because that is what you are looking for.*0442

*For example, f(x) = 2x + 7; if you wanted to find f ^{-1}(x), step 1 is to write y instead of f(x).*0446

*1, 2, 3, 4 steps: the first step is to write y instead of f(x).*0459

*The second step is to interchange x and y: I am going to change that to an x; I am going to change that to a y.*0468

*The third step is to solve for y: I am going to subtract 7 from both sides--I am trying to isolate y.*0478

*Now, I need to divide both sides by 2.*0487

*I have y isolated, but I am just going to rewrite this in a more standard form, with the isolated variable on the left side of the equation.*0495

*So, I have y = (x - 7)/2, and now I am going to write it as f ^{-1}(x), instead of using a y here.*0502

*I have found the inverse of f(x) by following these four steps.*0518

*One more example of constructing an inverse function: given f(x) = 5x - 3, we are going to find f ^{-1}(x).*0528

*Step 1: Change f(x) to y--let's rewrite this over here and follow each step down.*0542

*So, step 1--I am going to change this to y = 5x - 3.*0557

*Step 2: Interchange x and y--I am going to make this an x; I am going to make this a y.*0562

*Then, solve for y.*0579

*This was step 1; step 2; solving for y--I am going to add 3 to both sides;*0586

*then I need to divide both sides by 5; so I get (x + 3)/5 = y; and I am just going to rewrite that with the y on the left, in standard form here.*0595

*OK, next, replace y with f ^{-1}(x)--just change the notation.*0609

*This is going to be f ^{-1}(x) = (x + 3)/5.*0626

*So, I found the inverse of this function by using this four-step procedure.*0635

*Inverses and compositions: in the last lesson, we talked about composition functions.*0644

*And now, we are going to discuss how this relates to inverses, and how it can help you to determine if two functions are inverses of each other.*0652

*The functions f and g are inverses if and only if f composed with g of x equals x, and g composed with f of x equals x, for all x.*0663

*So, if I put a number in here (2), and I evaluate it for the composite function, I will get that same value back.*0678

*If I use 2, I will get 2 back; if I use 50, I will get 50 back; and the same when I look at g composed with f.*0691

*If I evaluate that for a particular value, 7, I will get 7 back.*0699

*Let's go ahead and take an example to illustrate this.*0707

*Given f(x) = 6x - 1, and g(x) = (x + 1)/6, are they inverses?*0710

*I can use this to evaluate whether or not two functions are inverses of each other.*0725

*And if they are inverses, then when I take f composed with g of x, I should get x back.*0732

*Let's look at what f composed with g of x is: recall that is the same as saying f(g(x)).*0743

*Replace this with g(x): well, g(x) is (x + 1)/6; f of this can be found by looking at the function and replacing x with this algebraic expression, (x + 1)/6.*0753

*So, f((x + 1)/6) is 6 times (x + 1)/6, minus 1.*0771

*And here, these 6's end up cancelling out; so that leaves me with x + 1 - 1 = x.*0788

*So, f composed with g of x does equal x; that is half of what I have to figure out.*0802

*Now, does g composed with f of x equal x?*0810

*Well, g composed with f of x equals g(f(x)); well, that is g(6x - 1), so I need to evaluate g for this algebraic expression.*0816

*Well, g is (x + 1)/6, so I need to substitute 6x - 1 right here, and then I need to add 1 to that value and divide it by 6.*0835

*I am going to remove these parentheses to get (6x - 1 + 1)/6.*0850

*The 1's cancel each other out to leave 6x/6; the 6's cancel to get x; so, g composed with f of x equals x.*0857

*And f composed with g of x is x, so are they inverses? Yes, f(x) and g(x) are inverses of each other.*0871

*So, this fact right here allows us to determine if two functions are inverses of each other or not.*0879

*OK, the first example: find the inverses of the relation (I am going to call this relation R, and I am asked to find the inverse of R).*0890

*Remember that, in the inverse, I flip around the two values; these domain values and the range values are going to be reversed.*0901

*That is going to give me {(3,1),(4,2),(7,3),(8,1),(9,4),(6,3)}; and then, I am just double-checking that they are all correctly reversed; and they are.*0913

*This is the inverse of the relation R.*0934

*Next, find the inverse of g(x) = 2x + 4.*0940

*All right, the first step is to change g(x) to y; that is going to give me y = 2x + 4.*0946

*Second, interchange x and y: OK, this is going to become x; this will become y.*0959

*The third step: solve for y--I am going to subtract both sides by 4: x - 4 = 2y.*0973

*Next, I am going to divide both sides by 2: (x - 4)/2 = y.*0986

*I have isolated y; I am just going to rewrite this with y on the left.*0993

*And then, I am going to replace y, or change y to the notation g ^{-1}(x).*0998

*I found that the inverse of this function is (x - 4)/2 by following these four steps.*1016

*Example 3: Find the inverse of h(x) = 2/3x - 4.*1027

*Again, here it is h(x), so change h(x) to y; that is going to give me y = 2/3x - 4.*1032

*The second step: interchange x and y--this is going to give me x = 2/3y - 4.*1045

*Next, solve for y: I need to add 4 to both sides, and then, after than, multiply both sides by 3/2.*1059

*And that is going to give me 3/2(x + 4) = y; multiplying this out, that is going to give me 3/2x + 3/2(4) = y.*1076

*3/2x...and this is 3 times 4 is 12, divided by 2; that becomes 6.*1094

*Finally, I am going to change y to h ^{-1}(x).*1102

*I am going to do that at the same time as I am going to go ahead and put the y on the left side, and this becomes h ^{-1}(x) = 3/2x + 6.*1111

*So, the inverse of the function given is h ^{-1}(x) = 3/2x + 6--again, I did this by following the four-step procedure.*1122

*Given f(x) and g(x), we are asked to determine if f and g are inverses of each other.*1137

*So, are f and g inverses of each other?*1145

*Now, recall that two functions are inverses of each other if and only if this scenario holds up.*1148

*If f composed with g of x equals x, and g composed with f of x equals x, then the functions are inverses.*1156

*OK, so let's go ahead and find f composed with g of x, which equals f(g(x)).*1182

*OK, f(g(x))--well, that is f of this expression--f(1/5(x + 8)).*1194

*Well, f(x) is 5x - 8, so f of this expression would be 5; then, replacing x with this, that is 1/5(x + 8) - 8.*1207

*OK, let's get rid of these parentheses and do some multiplying.*1224

*This is 5 times 1/5(x + 8) - 8; these 5's cancel, and I am just left with a 1 here; so that is just (x + 8) - 8 = x + 8 - 8 = x.*1228

*So, we have our first part: f composed with g of x is x; now, let's see if it works out for g composed with f.*1250

*Well, recall that that is going to be the same as g(f(x)).*1262

*Up here, f(x) is 5x - 8; so g(5x - 8) is what we are looking for; I need to evaluate g for this expression, this function, when x is 5x - 8.*1270

*So, that is 1/5...of (5x - 8); and then I am going to add 8 to that.*1284

*So, here I have 1/5...now let's get rid of some of these parentheses inside to simplify...5x - 8 + 8, getting rid of these...*1312

*That is 1/5, and then I have a negative 8 and a positive 8, so those become 0; so it is just 1/5 times 5x.*1325

*The 5's cancel out, so that equals x; so g composed with f of x is also x.*1337

*So, are these inverses of each other? Yes, f(x) and g(x) are inverses of each other, because these two hold true.*1346

*f composed with g of x is x, and g composed with f of x is also x.*1364

*That concludes this lesson of Educator.com; thanks for visiting!*1370

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