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| 12 Theory slides |
| 7 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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))=xandf-1(f(x))=x
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
Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|
f(f-1(x))=?x | 2f-1(x)−3=?x | 2(2x+3)−3=?x | x=x ✓ |
f-1(f(x))=?x | 2f(x)+3=?x | 22x−3+3=?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.
Kriz enjoys playing video games with their friends. For their birthday they received a copy of Mathleaks: The Adventure,
a math-based video game.
Graph:
Are f and g Inverse Functions? Yes.
Graph both functions on the same coordinate plane and see if they are each other's reflection across the line y=x.
Since f and g are linear functions written in slope-intercept form, they can both be graphed using their slope and y-intercept.
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.
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!
g(x)=51x−2
Distribute 5
Associative Property of Multiplication
5⋅5a=a
Multiply
Add terms
Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|
f(g(x)) | f(g(x))=5g(x)+10 | f(g(x))=5(51x−2)+10 | f(g(x))=x |
g(f(x)) | g(f(x))=51f(x)−2 | g(f(x))=51(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.
Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|
h(k(x)) | h(k(x))=21k(x)+6 | h(k(x))=21(2x−6)+6 | h(k(x))=x+3 |
k(h(x)) | k(h(x))=2h(x)−6 | k(h(x))=2(21x+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.
For the functions f and g, use a composition to determine whether they are inverse functions.
LHS⋅3=RHS⋅3
LHS+1=RHS+1
LHS/2=RHS/2
Rearrange equation
Zosia lives in Honolulu, Hawaii.
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).
To find the inverse of the function, there are four steps to follow.
These steps will be done one at a time.
LHS−24=RHS−24
LHS⋅(-43)=RHS⋅(-43)
Distribute (-43)
Commutative Property of Multiplication
a(-b)=-a⋅b
a⋅cb=ca⋅b
Calculate quotient
a−(-b)=a+b
Rearrange equation
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.
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).
To find the inverse of the function, there are four steps to follow.
These steps will be done one at a time.
Determine which of the diagrams, if any, show the graphs of two inverse functions.
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 ✓
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.
Find the inverse of the given functions.
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
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.