6.6A Functions and Their Inverses
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities, graph equations and/or inequalities on coordinate axes with labels and
scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations or
inequalities, and interpret solutions as viable or nonviable options in a modeling
situation.
For the Board: You will be able to determine whether the inverse of a function is a function, and write
rules for the inverses of functions.
Anticipatory Set:
If the inverse of the function is a function, then the inverse of the function f(x) is written f -1(x).
This does not represent a reciprocal.
Input
3
9
function
f(x) = x + 6
f-1(x) = x – 6
output
9
3
To find the inverse function use inverse operations.
Addition and subtraction are inverses.
Multiplication and division are inverses.
Powers and roots are inverses.
Open the book to page 243 and read example 2.
Example: Use inverse operations to write the inverse of f(x).
a. f(x) = x – 3
f-1(x) = x + 3
b. f(x) = 4x
f-1(x) = x /4
c. f(x) = x3
f-1(x) = 3 x
White Board Activity:
Practice: Use inverse operations to write the inverse of each function.
a. f(x) = x/3
b. f(x) = x + 2
f-1(x) = 3x
f-1(x) = x – 2
With multi-step operations inverse both the order of the operations and the operations themselves.
This can be confusing, so there is also an algebraic method of accomplishing the same result.
Open the book to page 243 and read example 3.
Example: Use inverse operations to write the inverse of f(x) = 3(x – 7).
Think order of evaluation: subtract 7 then multiply by 3.
Now reverse the order and the operation: divide by 3 then add 7.
x
f-1(x) = + 7
3
White board activity:
Practice: Use inverse operations to write the inverse of f(x) = 5x – 7.
Think evaluate: multiply by 5, then subtract 7.
Now reverse the order and the operation: add 7, then divide by 5.
x7
f-1(x) =
5
Instruction:
Algebraic Method:
1. replace f(x) with y.
2. interchange the x and y.
3. solve for y.
4. replace the y with f-1(x).
Example: Use algebra to write the inverse of f(x) = 4(x – 3)3 + 5.
y = 4(x – 3)3 + 5
x = 4(y – 3)3 + 5
x – 5 = 4(y – 3)3
x 5
x 5
x 5
3
3
y3
 y3
3
 y  3
4
4
4
White Board Activity:
Practice: Use algebra to write the inverse of f(x) = 3 x  1  2 .
y = 3 x 1  2
x  3 y 1  2
x  2  3 y 1
(x + 2)3 = y + 1
y = (x + 2)3 – 1
f-1(x) = (x + 2)3 - 1
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 245 – 246 prob. 4 – 13, 20 – 25.
Text: pgs. 453 – 454 prob. 24 – 28.
For a Grade:
Text: pgs. 245 – 246 prob. 20, 22, 24; pgs. 453 – 454 prob. 24.
f 1 ( x)  3
x 5
3
4