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Onto Onto OnOntoOnSupposeOntototo f : A —~ B is a function. We call f onto if the range of f equals Suppose f : A B is a function. We call f onto if the range of f equals B SuppSupposeSuppSupposeoseose ff f:: AA: A —~BBBisisisa a function.a function.function.W WeWWeee call callcallff fontoontoontoifififthe thetherange rangerange ofofofff fequalsequalsequals ! BBBInB.... other words, fis onto if every object in the target has at least one object fromInInInInother theotherotherotherdomainw words,wwords,words,ords,ords,assignedff fisisisison ontoonontotototoif ifififev everyiteveveryerybyeryf.ob objectobobjectjjectect in ininthe thethetarget targettargethas hashasat atatleast leastleast oneoneone objectobobobjectjectject fromfromfromfromfromthe thethethedomai domaindomaidomaidomainnn assigned assignedassignedto totoi itiittitb bybyybyff..f. Examples. Below is the graph of f 1W —> 1W where f(x) = z2. Using 2 Examples. Below is the graph of f : where f(x) = x22. Using techniquesExamExamples.ExamExamples.plples.es.learnedBeloBelowBeloBelowwwinis istheisthe thethechaptergraph graphgraphof of“Introofff f:: RRto1W Graphs”,—>RR1Wwherewherewhereweff(can(f(x)xx)=)) ==see=xxthatz2... UsingUsingUsingthe R ! R rangetectechniquestectectechniqueshniqueshniqueshniquesof f islearned learnedlearned[0,learnedoo). in ininTheinthe thethetargetc chapterchapterhapterchapterof f“In “Intro“Inis“Introtro1W,troandto totoGraphs”, Graphs”,[0,Graphs”,oo) $ 1W wewweweeso cancanfcanis seeseenotsee thatthatonto.that thethethe range of f is [0, ). The target of f is , and [0, ) = so f is not onto. rangerangerangeof ofoffffisisis[0 [0,[0,, oo).).). The TheThetarget targettargetof ofofff fisisisRR,1W,,and[0, andandand[0[0,[0,, oo))) ==$RR1Wsososoff fisisis notnotnot onto.onononto.to.to. ⇥1⇥ ⇥1⇥ ⇤6⇤⇤

Below is the graph of g 1W —~ 1W where g(x) = x3. The function g has the Below is the graph of g R —~ 1W where gQr) = 3x3. Ally horizontal line that set BelowBelo1W forwits isis thetherange. graphgraphThis ofof gequalsg :: R theRtargetwherewhere ofgg((xxg,)=) so= xxg33..is TheTheonto. functionfunction gg hashas thethe BeloBelocouldBelowww beisisisthethedrawnthegraphgraphgraphwouldofofofggintersectg:: RR1W —~RR1Wthewherewherewheregraphgg((g(x)xx)of) ==g=xxinx3... atTheTheThemostfunctionfunctionfunctionone point,gg ghashashassothethetheg is set for its range. This equals!the target of g, so g is onto. setsetsetone-to-oneRRR1Wforforforits itsitsrange. range.range.This ThisThisequals equalsequalsthe thethetarget targettargetof ofofgg,,g,so sosogg gisisison onto.ononto.to.to.

Onto* * * * * * * * * * * * * ***************** **** *** *** *** *** 67*** *** *** *** *** *** *** Suppose f : A —~ B is a function.9167We call f onto if the range of f equals 676767 B In other words, f is onto if every object in the target has at least one object from the domain assigned to it by f.

Examples. Below is the graph of f 1W —> 1W where f(x) = z2. Using techniques learned in the chapter “Intro to Graphs”, we can see that the range of f is [0, oo). The target of f is 1W, and [0, oo) $ 1W so f is not onto.

Below is the graph of g 1W —~ 1W where g(x) = x3. The function g has the set 1W for its range. This equals the target of g, so g is onto.

* * * * * * * * * * * * * 67 What an inverse function is Suppose f : A B is a function. A function g : B A is called the inverse function of!f if f g = id and g f = id. ! 1 If g is the inverse function of f, then we often rename g as f .

Examples.

Let f : R R be the function deﬁned by f(x)=x +3,andlet • ! g : R R be the function deﬁned by g(x)=x 3. Then !

f g(x)=f(g(x)) = f(x 3) = (x 3) + 3 = x

Because f g(x)=x and id(x)=x, these are the same function. In symbols, f g = id. Similarly

g f(x)=g(f(x)) = g(x +3)=(x +3) 3=x so g f = id. Therefore, g is the inverse function of f, so we can rename g 1 1 as f , which means that f (x)=x 3. Let f : R R be the function deﬁned by f(x)=2x +2,andlet • ! g : R R be the function deﬁned by g(x)=1x 1. Then ! 2

1 1 f g(x)=f(g(x)) = f x 1 =2 x 1 +2=x 2 2 ⇣ ⌘ ⇣ ⌘ Similarly

1 g f(x)=g(f(x)) = g(2x +2)= 2x +2 1=x 2 ⇣ ⌘ 1 1 Therefore, g is the inverse function of f, which means that f (x)=2x 1. 92 The Inverse of an inverse is the original 1 1 1 If f is the inverse of f, then f f = id and f f = id. We can see 1 from the deﬁnition of inverse functions above, that f is the inverse of f . 1 1 That is (f ) = f. Inverse functions “reverse the assignment” The deﬁnition of an inverse function is given above, but the essence of an inverse function is that it reverses the assignment dictated by the original 1 function. If f assigns a to b, then f will assign b to a. Here’s why: 1 If f(a)=b, then we can apply f to both sides of the equation to obtain 1 1 the new equation f (f(a)) = f (b). The left side of the previous equation 1 1 1 involves function composition, f (f(a)) = f f(a), and f f = id, so 1 we are left with f (b)=id(a)=a. 1 The above paragraph can be summarized as “If f(a)=b, then f (b)=a.”

Examples.

1 If f(3) = 4, then 3 = f (4). • 1 If f( 2) = 16, then 2=f (16). • 1 If f(x +7)= 1, then x +7=f ( 1). • 1 If f (0) = 4, then 0 = f( 4). • 1 2 2 If f (x 3x +5)=3,thenx 3x +5=f(3). • In the 5 examples above, we “erased” a function from the left side of the equation by applying its inverse function to the right side of the equation.

When a function has an inverse A function has an inverse exactly when it is both one-to-one and onto. This will be explained in more detail during lecture.

************* 93 Using inverse functions Inverse functions are useful in that they allow you to “undo” a function. Below are some rather abstract (though important) examples. As the semes- ter continues, we’ll see some more concrete examples.

Examples. Suppose there is an object in the domain of a function f,andthat this object• is named a. Suppose that you know f(a)=15. 1 1 If f has an inverse function, f , and you happen to know that f (15) = 3, 1 then you can solve for a as follows: f(a)=15impliesthata = f (15). Thus, a =3. If b is an object of the domain of g, g has an inverse, g(b)=6,and 1 • g (6) = 2, then 1 b = g (6) = 2 1 Suppose f(x +3)=2.Iff has an inverse, and f (2) = 7, then • 1 x +3=f (2) = 7 so x =7 3=4

*************

The Graph of an inverse If f is an invertible function (that means if f has an inverse function), and if you know what the graph of f looks like, then you can draw the graph of 1 f . 1 If (a, b)isapointinthegraphoff(x), then f(a)=b. Hence, f (b)=a. 1 1 That means f assigns b to a, so (b, a)isapointinthegraphoff (x). Geometrically, if you switch all the ﬁrst and second coordinates of points in R2, the result is to ﬂip R2 over the “x = y line”.

94 New How points in graph of f(x) visual effect NewNewfunction HoHowwbecomep pointsoints pointsin ingraph graphof ofnew of ff((graphxx)) visualvisuale e↵ectect functionfunction bbecomeecome p pointsoints of ofnew newgraph graph

f’(x) (a, b) (b, a) flip over the “x = y line” 1 f 1(x) (a, b) (b, a) ﬂip over the “x = y line” f (x) (a, b) (b, a) ﬂip over the “x = y line” ⇤⇥7!

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71 95 How to ﬁnd an inverse If you know that f is an invertible function, and you have an equation for 1 f(x), then you can ﬁnd the equation for f in three steps. Step 1 is to replace f(x)withthelettery. Step 2 is to use algebra to solve for x. 1 Step 3 is to replace x with f (y). After using these three steps, you’ll have an equation for the function 1 f (y).

Examples. Find the inverse of f(x)=x +5. •

Step 1. y = x +5

Step 2. x = y 5 1 Step 3. f (y)=y 5 2x Find the inverse of g(x)=x 1. •

2x Step 1. y = x 1 y Step 2. x = y 2 1 y Step 3. g (y)=y 2 Make sure that you are comfortable with the algebra required to carry out step 2 in the above problem. You will be expected to perform similar algebra on future exams. 1 1 You should also be able to check that g g = id and that g g = id.

96 Exercises

In #1-6, g is an invertible function.

1 1.) If g(2) = 3, what is g (3)?

1 2.) If g(7) = 2, what is g ( 2)? 1 3.) If g( 10) = 5, what is g (5)? 1 4.) If g (6) = 8, what is g(8)?

1 5.) If g (0) = 9, what is g(9)?

1 6.) If g (4) = 13, what is g(13)?

For #7-12, solve for x. Use that f is an invertible function and that 1 f (1) = 2 1 f (2) = 3 1 f (3) = 2 1 f (4) = 5 1 f (5) = 7 1 f (6) = 8 1 f (7) = 3 1 f (8) = 1 1 f (9) = 4

1 Remember that you can “erase” f by applying f to the other side of the equation.

7.) f(x +2)=5 8.) f(3x 4) = 3 9.) f( 5x)=1 1 5 10.) f( 2 x)=2 11.) f(x)=8 12.) f(x 1)=3 97 EachEacEachh ofofof thethethe functionsfunctionsfunctions givengivgivenen ininin #13-18#13-18#13-18 isisis invertible.ininvvertible.ertible. FindFindFind thethethe equationsequationsequations forforfor theirtheirtheir inverseininvEachverseerse functions.functions.offunctions.the functions given in #13-18 is invertible. Find the equations for their inverse functions. 13.)13.)13.)fff((x(xx)=3))==3x3xx+2++22 13.) f(z) = 3x + 2 14.)14.)14.)gg(g(x(xx)=))== xxx+5++55 14.) 1g(z)1 = —x + 5 15.)15.)hh((xx)=) = x1 15.) h(x) =x x

— 15.) h(z)xx 2J 16.)16.)16.)fff((x(xx)=))== x x1 x x1 1 2x2x+3+3 17.)17.)gg((xx)=) = 2xx+3 17.) g(x) = x x 2x+3 17.) g(x)xx = x 18.)18.)18.)hhh((x(xx)=))== 4 xx 4 4x x 18.)h(x)=~ 19.) Below is the graph of f : (0, ). Does f have an inverse? 19.)19.) BelowBelow isis thethe graphgraph ofofff: :RRR (0(0, , ).). DoesDoesffhavehave anan inverse?inverse? !⇥⇥ 1⇤⇤ 19.) Below is the graph of f R —÷ [0, oo). Does f have an inverse?

20.) Below is the graph of g : (0, ) . Does g have an inverse? 20.)20.) BelowBelow isis thethe graphgraph ofofgg:(0: (0, , ) ) RRR.. DoesDoesgghavehave anan inverse?inverse? 1⇤⇤ !⇥⇥ 20.) Below is the graph of g : [0, oo) —÷ 1W. Does g have an inverse?

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