Function Machines and Inverse Operations in Mathematics
Explore the concept of function machines with multiple operations, solve missing values, and learn about inverse operations to reverse functions. Practice writing inverse functions and draw function machines to understand the process better.
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Presentation Transcript
Sometimes a function has more than one operation involved. On your whiteboards use function notation to represent these function machines, with ? as the input. ? ? = 2? + 3 ? ? = 2(? + 3) ? =? 5 8 ? ? =? 8 5 ? ? = 7(?2+ 11)
? ? = 2? + 3 ? ? = 2(? + 3) ? =? 5 8 ? ? =? 8 5 ? ? = 7(?2+ 11)
Work out the missing values How did you calculate the missing input B?
To reverse the process of a function (go from the output to the input) inverse operations are used. ? ? The inverse function here would be written like this: ? 1? = 9 ? 7
On your whiteboards: Add 7 Divide by 9 9 7 What are the missing inverse operations? How do we write the inverse function? ? 1? = 9? 7
On your whiteboards: 5 +1 2 What are the missing inverse operations? How do we write the inverse function? ? 1? =? + 1 5 2
On your whiteboards: ? ? What are the missing inverse operations? How do we write the inverse function? ? 2 5 ? 1? =
In your books: Draw the function machines for each function and use it to work out the inverse function ? ? = 5? + 2 ? = 5? + 2 ? ? = ? + 2 ? = 5(? 2)2 ? ? = 5(? 2) ? ? = ? 2 ? =2? + 5 ? ? = 2? + 5 ? ? = 2? 3 ? ? =? ? 2 5 ? ? =? 2 5 ? = 3 2 ? =?2 3+ 5 ? ? = ?2 ? ? = ?2+ 5
Mark and correct your work ?2 2 5 ? 2 5 ? 1? = ? 2 1? = ? 1? = ? 1? = ? + 2 ? 5+ 2 ? 5+ 2 1? = ? 1? = ? 1? =? 3? 5 2 ? 5 5 2 1? = ? 1? = ? 1? = 2? 1? = 2(? ? 1? = 2 ? + 5 3+ 5) ? 1? = 1? = ? 1? = ? 5 3(? 5) ?
What is the inverse function? ? + 1 ? 2 ? ? = Drawing a function machine is not always possible
Worked Example If ? ? =?+1 ? 2, find ? 1(?). STEP 1: Write the output ?(?) as ? ? =? + 1 ? 2 This is purely for convenience. ?? 2? = ? + 1 STEP 2: Get the input in terms of the output (make ? the subject). ?? ? = 1 + 2? ? This is because the inverse function is the reverse process, i.e. finding the input ? in terms of the output ?. ? ? 1 = 1 + 2? STEP 3: Swap ? back for ? and ? back for ? 1? . ? =1 + 2? ? 1 This is because the input to a function is generally written as ? rather than ?. ? 1? =1 + 2? ? 1
Check Your Understanding Your Turn ? If ? ? =2?+1 If ? ? = 2? 1, find ? 1(?). 3, find ? 1(?). ? ? =?? + ? ? = ?? ? ? ??? ? = ? ?? = ?? + ? ??? ? = ? ? ?? ? = ?? ? ? ?? ? = ? ? =?? ? ? ? ? = ?? ? ? ?? =?? ? ? ? ? ?? = ?? ?
In your books Find ? 1(?) for the following functions. Find ? 1(?) for the following functions. ? ? + 3 ? ? =? 2 ? ? ? =2? 1 ? 1 1 ? 3? + 1 3? 3 + 2? 1 3 a ?? a b c ? ?? = ? ? = ? ? = 5?? ?? =? ? ? ? ? ? ? b ? ?? = ? ? ? ? ? = 1 + ? ? ? = 6? 4? ?? =?+? ? ? =?+7 ? ?? = ? d ? c ? ?? =? ? 3? ?? = ?? ? ? ? = 5 ? + 1? ?? = ? ? = 10 3?? ?? =?? ? e ? ? ? ? ?? + ? ?? ? ?? ? ? ? ? d f ? ?? = ? ? = ? e 2 ? ?? = ? ? = 2? ? ? = ? 1 ? Find ? 1? ? = ? ?
