3. Inverse of Linear Functions
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Chapter 3
3.
Inverse of Linear Functions
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Why
Understanding the inverse of a function is crucial in mathematics. The lesson delves into the concept of function inverses, particularly focusing on linear functions. It emphasizes the idea that every function has an inverse relation. If this inverse relation also qualifies as a function, it is called an
inverse function. The lesson provides a step-by-step guide on how to determine if two functions are inverses of each other, both algebraically and graphically. It highlights the significance of the line y=x, as the graphs of two inverse functions reflect across this line. The inverse essentially
undoesthe function, reversing its operations. This knowledge is pivotal for various applications, including real-world scenarios like currency conversion.
Lesson Settings & Tools
| | 12 Theory slides |
| | 7 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Inverse of Linear Functions
Slide of 12
Given a function f, sometimes it is possible to find another function that undoes f. This particular function is called the inverse function of f. This lesson will discuss how to verify whether two functions are inverse and will explain the procedure to find the inverse of a linear function.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Undoing a Function
Consider a function f that takes any real number and returns a number greater by two units.
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Discussion
Inverse Relation
An inverse relation is a relation that interchanges the input and output values of a specific relation. In other words, it is the relation that undoes a certain relation. Consider the mapping diagram that shows a relation p that takes elements from set A and returns elements from set B.
The relation that undoes p is q. In this case, q takes elements from set B and returns elements from set A. Relations can also be represented by a collection of ordered pairs. Here, the inverse of a relation is obtained by interchanging the elements of the ordered pairs of the original relation. Relationp {( - 1, C),( 0, E),( 5, Z),( 10, W),( 20, Q)} [1em] Inverse Relationq {( C, - 1),( E, 0),( Z, 5),( W, 10),( Q, 20)} Graphically, the inverse of a relation is a reflection across the line y=x.
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Discussion
Inverse of a Function
Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function f is another function f^(- 1) such that they undo each other.
f(f^(- 1)(x))=x and f^(- 1)(f(x))=x
Therefore, f and f^(-1) are inverses of each other. Also, if x is the input of a function f and y its corresponding output, then y is the input of f^(- 1) and x its corresponding output.
f(x)=y ⇔ f^(- 1)(y)=x
Consider a function f and its inverse f^(- 1). f(x)=2x-3 and f^(-1)(x)= x+32
Why
Showing That the Functions Are Inverse.The functions f(x)=2x-3 and f^(-1)(x)= x+32 will be shown to undo each other. To do so, it needs to be proven that f(f^(- 1)(x))=x and that f^(- 1)(f(x))=x. To start, the first equality will be proven. First, the definition of f will be used.
f(f^(- 1)(x))? =x ⇕ 2f^(- 1)(x)-3? =x
Now, in the above equation, x+32 will be substituted for f^(- 1)(x).
2f^(- 1)(x)-3? =x
Substitute
f^(- 1)(x)= x+3/2
2( x+3/2)-3? =x
▼
Simplify left-hand side
DenomMultFracToNumber
2 * a/2= a
(x+3)-3? =x
RemovePar
Remove parentheses
x+3-3? =x
SubTerm
Subtract term
x=x ✓
A similar procedure can be performed to show that f^(- 1)(f(x))=x.
| Definition of First Function | Substitute Second Function | Simplify | |
|---|---|---|---|
| f(f^(- 1)(x))? =x | 2f^(- 1)(x)-3? =x | 2( x+3/2)-3? =x | x=x ✓ |
| f^(- 1)(f(x))? =x | f(x)+3/2? =x | 2x-3+3/2? =x | x=x ✓ |
Therefore, f and f^(- 1) undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of f^(- 1) are the reversed points on the graph of f.
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Example
Using Graphs to Determine if Two Functions Are Inverse Functions
Kriz enjoys playing video games with their friends. For their birthday they received a copy of Mathleaks: The Adventure,
a math-based video game.
Kriz is very close to beating this level. They need to graph the functions given below and determine whether they are inverse functions. f(x)=4x+5 and g(x)=1/4x-5/4 Help Kriz beat the level and get one step closer beating the game by mastering math!
Answer
Graph:
Are f and g Inverse Functions? Yes.
Hint
Graph both functions on the same coordinate plane and see if they are each other's reflection across the line y=x.
Solution
Since f and g are linear functions written in slope-intercept form, they can both be graphed using their slope and y-intercept.
Now, it can be seen whether the lines are each other's reflection across y=x.
The lines are indeed each other's reflection across y=x. Therefore, f and g are inverse functions.
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Pop Quiz
Determining if Two Functions Are Inverses by Looking at Their Graphs
It was discussed previously that if the graphs of two functions are each other's reflection across y=x, then they are inverse functions. Therefore, it can be determined if two functions are inverse just by looking at their graphs. Determine whether the following linear functions are inverse by looking at the lines.
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Example
Verifying Inverse Functions by Composition
If Paulina gets an A on a math test, her mother will buy her a new saxophone.
To get an A, Paulina must answer two questions correctly. Help her get the new saxophone!
a Given f(x)=5x+10 and g(x)= 15x-2, calculate f(g(x)) and g(f(x)), and determine whether f and g are inverse functions.
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b Given h(x)= 12x+6 and k(x)=2x-6, calculate h(k(x)) and k(h(x)), and determine whether h and k are inverse functions.
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Hint
a If f(g(x))=x and g(f(x))=x, then f and g are inverse functions.
b If h(k(x))=x and k(h(x))=x, then h and k are inverse functions.
Solution
a Consider the given functions.
f(x)=5x+10 and g(x)=1/5x-2 The composite function f(g(x)) will be found first. To do so, the definition of f will be used. f(g(x))=5g(x)+10 Next, g(x)= 15x-2 can be substituted in the above equation.
f(g(x))=5g(x)+10
Substitute
g(x)= 1/5x-2
f(g(x))=5( 1/5x-2)+10
▼
Simplify right-hand side
Distr
Distribute 5
f(g(x))=5(1/5x)-5(2)+10
AssociativePropMult
Associative Property of Multiplication
f(g(x))=5(1/5)x-5(2)+10
DenomMultFracToNumber
5 * a/5= a
f(g(x))=1x-5(2)+10
Multiply
Multiply
f(g(x))=x-10+10
AddTerms
Add terms
f(g(x))=x
By following the same procedure, an expression for g(f(x)) can be found.
| Definition of First Function | Substitute Second Function | Simplify | |
|---|---|---|---|
| f(g(x)) | f(g(x))=5g(x)+10 | f(g(x))=5( 1/5x-2)+10 | f(g(x))=x |
| g(f(x)) | g(f(x))=1/5f(x)-2 | g(f(x))=1/5( 5x+10)-2 | g(f(x))=x |
It has been found that both f(g(x)) and g(f(x)) are equal to x. Therefore, f and g are inverse functions.
b Consider the given functions.
h(x)=1/2x+6 and k(x)=2x-6 By following the same procedure as in Part A, the expressions for h(k(x)) and k(h(x)) will be found.
| Definition of First Function | Substitute Second Function | Simplify | |
|---|---|---|---|
| h(k(x)) | h(k(x))=1/2k(x)+6 | h(k(x))=1/2( 2x-6)+6 | h(k(x))=x+3 |
| k(h(x)) | k(h(x))=2h(x)-6 | k(h(x))=2( 1/2x+6)-6 | k(h(x))=x+6 |
It has been found that neither h(k(x)) nor k(h(x)) is equal to x. Therefore, h and k are not inverse functions.
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Pop Quiz
Determining if Two Functions Are Inverse by Composition
For the functions f and g, use a composition to determine whether they are inverse functions.
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Discussion
Finding the Inverse of a Function Algebraically
A function can be represented by a table of values, a graph, a mapping diagram, or a function rule, among other ways. Depending on how the function is presented, finding its inverse can be done in different ways. When a function rule is given, finding the inverse algebraically is advantageous. Consider the following example function. f(x)= 2x-1/3 There is a series of steps to follow in order to find the inverse function f^(- 1)(x).
1
Replace f(x) With y
expand_more To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule.
f(x)= 2x-1/3 → y= 2x-1/3
2
Switch x and y
expand_more Because the inverse of a function reverses x and y, the variables can be switched. Notice that every other piece in the function rule remains the same. y= 2 x-1/3 switch x= 2 y-1/3
3
Solve for y
expand_more Solve the resulting equation from the previous step for y. This will involve using the inverse operations.
x=2y-1/3
▼
Solve for y
MultEqn
LHS * 3=RHS* 3
3x=2y-1
AddEqn
LHS+1=RHS+1
3x+1=2y
DivEqn
.LHS /2.=.RHS /2.
3x+1/2=y
RearrangeEqn
Rearrange equation
y=3x+1/2
4
Replace y With f^(- 1)(x)
expand_more Just as f(x)=y shows the input-output relationship of f, so does f^(- 1)(x)=y. Therefore, replacing y with f^(- 1)(x) gives the rule for the inverse of f.
y=3x+1/2 → f^(- 1)(x)=3x+1/2
Notice that in f, the input is multiplied by 2, decreased by 1, and divided by 3. From the rule of f^(- 1), it can be seen that x undergoes the inverse of these operations in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.
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Example
Finding the Inverse of a Function
Zosia lives in Honolulu, Hawaii.
She is obsessed with math. While watching a surfer catch some gnarly waves, she came up with a function that models the distance of the surfer from the shore. f(x)=- 4/3x+24 Here, f(x) is the distance of the surfer from the shore after x seconds of riding a wave. Given the distance from shore, Zosia now wants to finds the function that gives the time, in seconds, the surfer has been riding a wave. To do so, she needs to find f^(- 1), the inverse of f. Help Zosia find this function! Write the answer in slope-intercept form.
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Hint
Start by replacing f(x) with y. After that, switch the x- and y-variables. Then, solve the obtained equation for y. Finally, replace y with f^(- 1)(x).
Solution
To find the inverse of the function, there are four steps to follow.
- Replace f(x) with y.
- Switch x and y.
- Solve the obtained equation for y.
- Replace y with f^(- 1)(x).
These steps will be done one at a time.
Step 1
Here, the variable y will be substituted for f(x) in the given function. f(x)=- 4/3x+24 → y=- 4/3x+24
Step 2
In this step, the variables x and y must be switched. y=- 4/3 x+24 switch x=- 4/3 y+24
Step 3
Here, the equation obtained in the previous step will be solved for y.
x=- 4/3y+24
▼
Solve for y
SubEqn
LHS-24=RHS-24
x-24=- 4/3y
MultEqn
LHS * (- 3/4)=RHS* (- 3/4)
(x-24)(- 3/4)=y
Distr
Distribute (- 3/4)
x(- 3/4)-24(- 3/4)=y
CommutativePropMult
Commutative Property of Multiplication
- 3/4x-24(- 3/4)=y
MultPosNeg
a(- b)=- a * b
- 3/4x-(- 24(3/4))=y
MoveLeftFacToNum
a*b/c= a* b/c
- 3/4x-(- 72/4)=y
CalcQuot
Calculate quotient
- 3/4x-(- 18)=y
SubNeg
a-(- b)=a+b
- 3/4x+18=y
RearrangeEqn
Rearrange equation
y=- 3/4x+18
Step 4
Finally, f^(- 1)(x) will be substituted for the y-variable in the equation obtained in the previous step. y=- 3/4x+18 ↓ f^(- 1)(x)=- 3/4x+18
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Pop Quiz
Finding the Inverse Function
Find the inverse of the given function. Write the answer in slope-intercept form — in the format y=mx+b. If the slope and y-intercept are not integers, express them as decimal numbers. If necessary, round them to two decimal places.
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Closure
Undoing a Function
In this lesson, it has been seen that the function that undoes a function f is its inverse f^(- 1). Consider the function given in the challenge presented at the beginning of the lesson.
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Hint
Start by replacing f(x) with y. After that, switch the x- and y-variables. Then, solve the obtained equation for y, and finally replace y with f^(- 1)(x).
Solution
To find the inverse of the function, there are four steps to follow.
- Replace f(x) with y.
- Switch x and y.
- Solve the obtained equation for y.
- Replace y with f^(- 1)(x).
These steps will be done one at a time.
Step 1
Here, the variable y will be substituted for f(x) in the given function. f(x)=x+2 substitute y=x+2
Step 2
In this step, the variables x and y must be switched. y= x+2 switch x= y+2
Step 3
Here, the equation obtained in the previous step will be solved for y. x=y+2 ⇔ y=x-2
Step 4
Finally, f^(- 1)(x) will be substituted for the y-variable in the equation obtained in the previous step. y=x-2 substitute f^(- 1)(x)=x-2 It has been found that the inverse function of f(x)=x+2, and therefore the function that undoes it, is f^(- 1)(x)=x-2
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Inverse of Linear Functions
Exercises
Determine which of the diagrams, if any, show the graphs of two inverse functions.
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Recall that the graphs of two inverse functions are each other's reflection across the line y=x. This means that the points on the graph of the inverse of a function are the reversed points on the graph of the function. To visualize this, let's consider the graphs of two inverse functions.
In the diagram, we see that the graphs of f(x) and its inverse f^(-1)(x) are each other's reflection across y=x. To see which of the given diagrams show the graphs of two inverse functions, we will reflect one of the lines across y=x and verify whether it is mapped onto the other line.
In diagrams B and C, the lines that are reflected across y=x are not mapped onto the other line of the diagrams. Therefore, these diagrams do not show the graphs of inverse functions. B * C * Conversely, in diagrams A and D, the lines that are reflected across y=x are mapped onto the other line of the diagrams. Therefore, diagrams A and D show graphs that are each other's reflections across the line y=x. These are the diagrams that show the graphs of inverse functions. A ✓ D ✓
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Solution
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{"type":"choice","form":{"alts":["f and g are inverse functions.","f and g are not inverse functions."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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To check whether f and g are inverse functions, we will find expressions for f(g(x)) and g(f(x)). If both compositions are equal to x, then f and g are inverse functions. Let's start by calculating f(g(x)). To do so, we need to evaluate f when x=g(x). f(x)=2x+5 ⇓ f( g(x))=2 g(x)+5 Now, we will use the definition of g and simplify the right-hand side of the equation. By doing this, we will write an expression for f(g(x)).
We found that f(g(x))=x. So far, this means that f and g can be inverse functions. By following the same procedure, let's now calculate g(f(x)). We can start by evaluating g when x=f(x). g(x)=1/2x-5/2 ⇓ g( f(x))=1/2 f(x)-5/2 We can now use the definition of f.
We also found that g(f(x))=x. Therefore, f and g are inverse functions!
Just like in Part A, we will calculate the composition of the given functions. Let's start with f(g(x)).
f(x)=2x+2 ⇓ f( g(x))=2 g(x)+2
We will now use the definition of g.
Because f(g(x)) is equal to 4x-2, which is not x, we can say that f and g are not inverse functions.
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Solution
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In order to find the inverse of the given function, we will start by replacing f(x) with y. f(x)=2x+7 → y=2x+7 Now, to algebraically determine the inverse of the given equation we switch the x and y variables. cc Given Equation & Inverse Equation [0.8em] y=2 x+7 & x=2 y+7 Finally, we will solve for y. The resulting equation will be the inverse of the given function.
We will write the inverse of the given function in function notation by replacing y with f^(- 1)(x) in our new equation. f^(- 1)(x)=1/2x-7/2
As we did before, we will start to find the inverse of the given function by replacing g(x) with y.
g(x)=1/3x-5/3 → y=1/3x-5/3
Then, to algebraically determine the inverse of the given equation, we exchange x and y and solve for y.
cc
Given Equation & Inverse Equation [1em]
y=1/3 x-5/3 & x=1/3 y-5/3
Finally, we will isolate y in the new equation to find the inverse of the given function.
Like we did in Part A, we will write the inverse of the given function in function notation by replacing y with g^(- 1)(x) in our new equation. g^(- 1)(x)=3x+5
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Solution
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The given function expresses euros in terms of US dollars. The inverse of this function expresses US dollars in terms of euros. Therefore, to find the number of US dollars that can be converted from 300 euro on this day, we will find the inverse of the given function. Let's do it!
We will now use the new formula to find the number of US dollars that can be bought with 300 euro. To do this, we will substitute 300 for E and evaluate the right-hand side of the equation.
We found that, on this certain day, 300 euro can buy about 342.11 US dollars.
In Part A, we found the formula that converts euro into US dollars. D=10 000/8769E We want to find the number of US dollars that can be bought with 500 euro on this particular day. To do this, we will substitute 500 for E and evaluate the right-hand side of the equation.
We found that, on this certain day, 500 euro can buy about 570.19 US dollars.
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Inverse of Linear Functions
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