Inverse Functions © Christine Crisp

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

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

Function Inverse Function x2 xa   + - reciprocate   - + Remember the inverse function performs the reverse effect Function Inverse Function x2 xa   + - reciprocate   - +  

Using the Reciprocal Function Ex.1 f(x)= find f–1 (x) 4 3 2 1 f(x) x To find the inverse we need a function which will change ½ back into 2 and ¼ back into 4 etc So the inverse of is f(x) = and f–1(x) =

Function Inverse Function x2 xa   + - reciprocate   - + Remember the inverse function performs the reverse effect Function Inverse Function x2 xa   + - reciprocate   - + reciprocate

x   x +3  square  x 2  f(x) f –1 (x)  -3  square root   2 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  +3  square  x 2  f(x) To find the inverse go backwards finding the inverse of each operation f –1 (x)  -3  square root   2  x so f –1 (x) =

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

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

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

x  Power   2 -5  f(x) Ex.3 f:x find f -1(x) List the operations in the order applied x  Power   2 -5  f(x) Go backwards finding the inverse of each operation  Power   2 +5  x f–1(x) f–1(x)    f –1 (x)=

Ex 4 Changing the Sign Ex.1 f:x 5 - x To change the sign of x multiply by –1 x  -1  +5  f(x) inverse of -1 is  -1 Which is the same as -1 f–1(x)   -1 -5  x f–1(x) =

Ex 5 x  -3  +4  f(x) inverse of -3 is  -3 The inverse is

Exercise Find the inverses of the following functions: 2. 3. 1. 4.

Solution: 1. So, Solution: 2. So, Solution: 3. So, Solution 4. So,

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