FunctionsInverse Functions

Team sports is a world of split-second decisions and fast-moving bodies. Teammates choose where to place a pass. Referees decide who gains possession of the ball. Fans follow the action from great distances. In all these cases, jersey colour is deciding factor in how these instantaneous decisions are made.

Let's take a look at the the Balligon Soccer League. Match each team to a jersey colour such that you have a function.

team mascots

Jersey Colours

Recall that the team mascots make up the domain range of the function,

while jersey colours make up the range. In function notation, we write this relationship as f()=.

Take a look at the end of season tournament brackets.

This idea of seeing jersey colour and immediately knowing what team the player belongs to is the opposite of fteam=jersey colour. In math, we call this backwards relationship an inverse. We use this idea when we balance equations: addition cancels subtraction, multiplication cancels division, etc. Specifically, we are thinking about an inverse function relationship.

The inverse function we are looking at identifies the team based on jersey colour. Take a look at the mapping for the function above moving from left to right.

Here we can see that the relation between colour and team is not a function. What would fix this relationship between fteam and so that f−1colour is a function?

Now f−1colour is a function. In fact, recall from Relations and Functions there are several types of functions. fteamis now a one-to-one one-to-many many-to-one many-to-many.

Let's look at this mapping relationship in a function machine. Choose a team to pass through the machine. The machine assigns the team to a colour, then reverses the assignment. In other words, the machine applied the function rule then the inverse function rule.

Inspiration

Every time you pass through both machines, you get out the team's name jersey color. Algebraically we write this as ff1x=x. We will look more closely at why this is important throughout the chapter.

Currency Conversion

International tourism reached over 4 million arrivals each day in 2019. With so much travel between countries, people are more aware of their spending power. That is, how far a user’scurrencyunite.g.dollar in user’shomecountry’scurrency goes in another currency. Let's explore this idea of spending power - choose a destination from the cities listed.

Cities and Currencies

Along your trip, you find many different markets take only cash. To prepare for your next trip to the market, you withdraw cash. The conversion function is destinationcurrency=conversionrate·homecurrency . Based on your budget you have about

user enters whatever amount they want

to spend on souvenirs, which means you need to withdraw

user must do this calculation and enter the amount +- .5

Take a stroll through the market and decide what you want to buy.

Figma

Souvenirs by city

Visual relationship

We can represent this relationship between the cost in destinationcurrency and homecurrency as (destinationcurrency, userhomecurrency). This relationship means destinationcurrency is the input output and userhome is the output input. The points fall on a line. Let's call this line f(destinationcurrency)=conversionratexdestinationcurrency.

Recall that earlier we used the equation destinationcurrency=conversionrate·homecurrency to calculate how much money to withdraw from the bank before our trip to the market. Notice that the this equation is strikingly similar to the f(x) on the graph.

In fact, the most important difference is that the input and output values are switched. We saw this relationship between functions above in the jersey example. Using this observation, we can see that destinationcurrency=conversionrate·homecurrency is the inverse function of f(destinationcurrency)=conversionratexdestinationcurrency. This means we can use the notation f−1homecurrency for the function representing the exchange from homecurrency to destinationcurrency.

The graphs of inverse functions have some interesting features. First, this idea of switching inout and output values helps us plot points for f−1homecurrency. Let's plot the souvenir points for the inverse function.

https://www.desmos.com/calculator/ff6489z9ns

https://www.desmos.com/calculator/tlscgnsyy3

https://www.desmos.com/calculator/ccbs3quxpu

https://www.desmos.com/calculator/mgmj8jbz26

There is another interesting feature of inverse functions, though it may not be immediately obvious. Let's make some measurements between lines to see if the interesting feature emerges.

Begin by measuring the vertical distance from souvenir1 on f−1homecurrencyto fdestinationcurrency. The distance is

fcostofsouvenir1∈destinationcurrency- f−1costofsouvenir1∈homecurrencyxxx. Similarly measure the horizontal distance between souvenir1 on fdestinationcurrency to f−1homecurrency.

This distance is the same as the vertical distance you measured! Make the same measurements for each of the souvenir points.

Notice that the intersection points of each of these measurements fall on their own line.

The equation of this line is y= m+

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which simplifies to y=x. Notice that if we were to fold the coordinate plane along the line y=x, fdestinationcurrency and f−1homecurrencywould lie on top of each other. The line y=x would remain in the same position during this fold.

In fact, all inverse functions are symmetric over the line y=x. The tools we used to arrive at this observation are the same that we can use to determine whether two functions are inverses. Let's take a closer look at this with a new graph shape called a parabola.

Parabolas

The parabola passes the vertical line test, so it is a function. We just observed that inverses are symmetric about the line y=x. Make a copy of the parabola and rotate it in the coordinate plane to make it a reflection of the original parabola over the line y=x. Reflection in this case means that the two parabolas are symmetric over y=x.

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The inverse function doesn’t pass the vertical line test. It isn’t actually a function! When we were looking at the jersey example, we changed the team to colour function so that each team was assign to only one colour. Can you apply that idea here?

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https://www.desmos.com/calculator/ywpamz642r

When we change the domain of the original parabola, we can make the inverse parabola a function. In fact, the domain must be restricted so that each input value is matched to only one output. This means that some functions have inverses and some functions don’t. Specifically, only one-to-one functions have inverse functions.

Algebraic Relationship

The forecast calls for a high of 27 degrees Celsius today. We missed the report that included the temperature in Fahrenheit, so we Google the conversion and find C(x)= . Find today's high in Fahrenheit.

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Substitute 27 in for C(X)

Use inverse operation on to move 9

Use inverse operation to move 5

Use inverse operation to move 32

x, representing the temperature in Fahrenheit, is isolated

Today's high in Fahrenheit

This is a familiar process for solving equations. We can use this same process to find inverse functions algebraically. Let's take a look at two key steps in solving the equations above.

Recall that we substituted 27 in for C(x) since it was our given temperature. Let's put y in for 27 to represent the output value in an ordered pair.

Recall from the jersey example above that passing a value through a function machine then its inverse function machine gives the same value in the end. We called this ff−1x=x. Let's verify that this works for C(x) and C−1x.

Let's determine whether these two functions are inverses.

The function machine indicates that f(x) and g(x) are not are inverse functions.

Let's find the inverse function of f(x).

We know have several ways of identifying and verifying inverse functions. Select your favourite method and describe why.

[option 1](inverting equations algebraically) [option 2](graphing using coordinate points and their inverses) [option 3](graphing using y=x as the line of symmetry) [option 4](mapping diagram)

Functions and inverse functions are fundamental ideas in coding and deciphering. Check out Codes and Ciphers to see these ideas in action!