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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.
Now, it can be seen whether the lines are each other's reflection across y=x.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.
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.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