2: Inverse Functions © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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Inverse Functions Suppose we want to find the value of y when x = 3 if We can easily see the answer is 10 but let’s write out the steps using a flow chart. We have To find y for any x, we have To find x for any y value, we reverse the process. The reverse function “undoes” the effect of the original and is called the inverse function. The notation for the inverse of is

Inverse Functions e.g. 1 For, the flow chart is Reversing the process: Finding an inverse The inverse function is Tip: A useful check on the working is to substitute any number into the original function and calculate y. Then substitute this new value into the inverse. It should give the original number. Notice that we start with x. Check: e.g. If

Inverse Functions FunctionInverse Function x2x2 xaxa   + - reciprocate Remember the inverse function performs the reverse effect   - +

Inverse Functions Using the Reciprocal Function Ex.1 f(x)= find f –1 (x) To find the inverse we need a function which will change ½ back into 2 and ¼ back into 4 etc f –1 (x) = So the inverse of is f(x) = and f –1 (x) = f(x)x 1 4

Inverse Functions FunctionInverse Function x2x2 xaxa   + - reciprocate Remember the inverse function performs the reverse effect   - + reciprocate

Inverse Functions Finding the inverse of a function Ex.1 f:x= 2(x+3) 2 find f –1 (x) List the operations in the order applied x  To find the inverse go backwards finding the inverse of each operation  x so f –1 (x) = Domain x  0 as you cannot  a negative number +3  square  x 2  f(x)   2  square root  -3f –1 (x)

Inverse Functions As the original x value is obtained the inverse function is correct The result can be checked by substitution so f(2) = substitute this value into the inverse function f -1 (x) f -1 (50) = f(x)= 2(x+3) 2 2(2+3) 2 = 50

Inverse Functions x  Ex.2 f:x  x  find f -1 (x) f(x)  x f –1 (x) List the operations in the order applied Go backwards finding the inverse of each operation 33 -4  reciprocate 22 +5   -5  2 2  reciprocate  + 4  3 3f –1 (x)

Inverse Functions Checking f(2) = Substitute x = 6 into f –1 (x) f –1 (6) This is the original x value. The result can be checked by substitution =2

Inverse FunctionsConsider Why are these the same? ANS: add up the fractions x xf   1 3 )( 1 An alternative Answer Cross and Smile

Inverse Functions Ex.2 f:x  x  find f -1 (x) f –1 (x) Done earlier Cross and Smile

Inverse Functions Changing the Sign Ex.1 f:x  5 - x To change the sign of x multiply by –1 x   -1  +5  f(x) f –1 (x)   -1  -5  x inverse of  -1 is f –1 (x) = Which is the same as  -1  -1

Inverse Functions Ex The inverse is x   -3  +4  f(x) inverse of  -3 is  -3

Inverse Functions The previous example was for The inverse was Suppose we form the compound function. Can you see why this is true for all functions that have an inverse? ANS: The inverse undoes what the function has done. f –1 (4 – 3x)

Inverse Functions The order in which we find the compound function of a function and its inverse makes no difference. For all functions which have an inverse,

Inverse Functions Exercise Find the inverses of the following functions:  4. See if you spot something special about the answer to this one. Also, for this, show

Inverse Functions So, Solution: 1.  Solution: 2. So, Solution: 3. So, Solution 4. So,

Inverse Functions Using Long Division to Find Inverses As x appears in 2 places it is impossible to go forwards and backwards using the order of operations. Do long division. So now x appears in only 1 place. Ex.1 f:x  x  -2 x + 2 x 1 x+2 -2 x+2 2 –

Inverse Functions f: x  xx f –1 (x) =    x  – 1 f –1 (x) = List the operations in the order applied Go backwards finding the inverse of each operation Simplify f –1 (x) +2  recip  -2  +1  f(x)

Inverse Functions SUMMARY To find an inverse function: Write the given function as a flow chart. Reverse all the steps of the flow chart.

Inverse Functions

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Inverse Functions is an example of a many-to-one function One-to-one and many-to-one functions is an example of a one-to-one function Consider the following graphs and

Inverse Functions e.g. 1 For, the flow chart is Reversing the process: Finding an inverse The inverse function is Notice that we start with x. Check:e.g. If

Inverse Functions The flow chart method of finding an inverse can be slow and it doesn’t always work so we’ll now use another method. e.g. 1 Find the inverse of Solution: Rearrange ( to find x ): Let y = the function: Swap x and y : So,

Inverse Functions or: e.g. 2 Find the inverse function of There are 2 ways to rearrange to find x : Solution: Let y = the function: Swap x and y: Either:

Inverse Functions So, for x x xf x xf    3 )(1 3 )( 11 or

Inverse Functions e.g. 3 Find the inverse of Solution: Rearrange: Multiply by x – 1 : Remove brackets : Collect x terms on one side: Remove the common factor: Swap x and y : Divide by ( y – 2) : So, Let y = the function:

Inverse Functions SUMMARY To find an inverse function: EITHER: Write the given function as a flow chart. Reverse all the steps of the flow chart. OR: Step 2: Rearrange ( to find x ) Step 1: Let y = the function Step 3: Swap x and y