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Jaime Foxx restricts the domain? (Produce an invertible function from a non-invertible function by restricting the domain.)

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Jaime Witherspoon
Lindsey Foxx
Jaelyn Nelson

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Jaime Foxx is trapped and needs your help! Jaime managed to send you a secret message. The message says how he is trapped and needs your math expertise to save him. The only way to save Jaime is if you decode a highly advanced inverse function. A mathematical kit, with steps to decode  the message, has been provided to you. Can you decode the message in time to save Jaime?

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Students will be able to produce an invertible function from a non-invertible function by restricting the domain in order to solve real world problems (e.g: metric conversions; decoding a message)

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30 minutes each class period for two weeks.

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F.BF.4-  Build new functions from existing functions

Find inverse functions

F.BF.4d- Produce an invertible function from a non-invertible function by restricting the domain.

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What is the end-product the students will produce? Who will use/care about the product?  Why will the product be meaningful to students?  How is technology integrated within this product? How will you assess the product?

Students will create a presentation explaining the steps they took to produce an invertible function from a non-invertible function. Students will focus their research on how to algebraically solve an invertible function and to use those skills to apply them to real-world applications such as decoding a message or converting metric units.

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 What help would you like to receive from Ms. Tunkara? 

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 What help would you like to receive from Ms. Tunkara? 

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How can I better simulate the problem we have given the students?

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Standard: F.BF.4d : produce an invertible function from a non-invertible function  by restricting the domain.

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Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if f takes a to b, then the inverse, ​​f⁻¹, start superscript, minus, 1, end superscript, must take b to a.

a->f->b->f⁻¹->a

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Now that we have discussed what an inverse function is, the notation used to represent inverse functions, one-to-one functions, and the Horizontal Line Test, we are ready to try and find an inverse function.

Here are the steps required to find the inverse function:

Step 1:  Determine if the function has an inverse. Is the function a one to one function? If the  function is a one to one function, go to step 2. If the function is not a one to one function, then say that the function does not have an inverse and stop.

Step 2:  Change f(x) to y.

Step 3:  Switch x and y.

Step 4:  Solve for y.

Step 5:  Change y back to f (x ).

By following these 5 steps we can find the inverse function. Make sure that you follow all 5 steps. Many  people will skip step 1 and just assume that the function has an inverse; however, not every function has an  inverse,  because not every function is a one-to-one function. Only functions that pass the Horizontal Line  Test are one-to-one functions and only one-to-one functions have an inverse.

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Symbol
f⁻¹(x)

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Meaning

The composition of the function and its inverse ( and vice versa) equals x

.(f*f⁻¹)(x)=(f⁻¹*f)(x)=x

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How to Find

The inverse of a relation in an equation is found by exchanging x- and -y variables in an equation and solve for x.

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How to check answer

Find the compositions f*f⁻¹(x) and f⁻¹*f(x). If both answers are x, then the functions are inverses.

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note

The symbol -1 in f^-1 is not an exponent

f⁻¹(x)≠1/f(x)

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Example 1: Find the inverse of the function f(x)=3x+6

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Example 1: Find the inverse of the function f(x)=3x+6

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Step 1: Change f(x)to Y                        y=3x+6

Step 2 : change each x variable to y. change each y-variables to x-variables                                                 x=3y+6
Step 3: Solve the equation for y, which means get y by itself                                  x=3y+6

x-6=3y

x/3 -6/3= 3y/3

x/3-2=y
Step 4 : write answer with y on the left and in inverse notation
y=x/3-2 -> f^-1=x/3-2

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1. To which intervals could we restrict the domain of f  to make it an invertible function? Choose all that apply
a. 1≤ x ≤10

b.−3≤ x ≤3

c. None of the above

d. −10≤ x ≤−6

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 Linear function: Find the inverse of g(x)=2x-5

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 Linear function: Find the inverse of g(x)=2x-5

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Cubic function: Find the inverse of h(x)=x ³+2

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What is the inverse of the following function?  f(x)=2x ²

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Direction: Find the inverse of each quadratic function. Show all your work in the space provided.

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Direction: Find the inverse of each quadratic function. Show all your work in the space provided.

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f(x)=x2⁻² for x<0

)f(x)=-x ²-2 for x ≥ 0

f(x)=x ²+6x+4 for x <-3

f(x)=(x-1)2 for x< 1

f(x)=-(x+1)² for x > -1

f(x)=x ²-2x-5 for x>1