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In this lesson, we will learn how to use compositions of functions to find and justify inverses of functions.

Q1:

Which of the following functions is NOT its own inverse?

Q2:

Suppose π ( π₯ ) = π π₯ + π is the inverse function to π ( π₯ ) = 2 β 3 π₯ . Determine π and π by considering the composition π ( π ( π₯ ) ) .

Q3:

Let π ( π₯ ) = 2 π₯ β 1 3 π₯ β 2 . Determine which of the following functions π is the inverse of π by checking whether π ( π ( π₯ ) ) = π₯ .

Q4:

Determine π and π so that π ( π₯ ) = π + π β π₯ is an inverse function to π ( π₯ ) = ( π₯ β 3 ) ο¨ by considering π ( π ( π₯ ) ) .

Q5:

Shown are the graphs of functions π and g from the set { 1 , 2 , 3 , 4 , 5 , 6 } into itself. Write the function β ( π₯ ) = π ( π ( π₯ ) ) as a list of ordered pairs. What can you conclude?

Q6:

Let π ( π₯ ) = 2 π₯ β 1 3 π₯ β 2 . Solve π¦ = π ( π₯ ) to find an expression for π ( π₯ ) β 1 .

Find π ( π₯ ) β 1 and π ( π₯ ) β 1 when π ( π₯ ) = 5 π₯ + 2 2 π₯ β 5 and π ( π₯ ) = 5 π₯ + 2 2 π₯ + 1 .

Let π ( π₯ ) = π π₯ + π π π₯ + π and π ( π₯ ) = π π₯ β π β π π₯ + π . By considering the simplification of π ( π ( π₯ ) ) , what, if any, is the relation between π and π ?

Let π ( π₯ ) = 2 π₯ + 1 3 π₯ β 2 , π ( π₯ ) = β 2 π₯ β 1 β 3 π₯ + 2 , π ( π₯ ) = 4 π₯ + 2 6 π₯ β 4 1 2 a n d . By considering the simplifications of π β π 1 and π β π 2 , what, if any, are the relations between these functions?

Suppose that π = π β 1 , where π ( π₯ ) = π π₯ + π π π₯ + π . What must be true of π , π , π , and π ?

Q7:

Which of the following is a necessary condition for a function π to be invertible?

Q8:

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