2.7+Inverse+Functions
Section 2.7 Inverse Functions
Key Concepts:
Inverses of relations
Verify inverse functions
Inverse functions and the horizontal line test
Finding the inverse of a function
Use the graph of a one-to-one function to graph its inverse function.
Find the inverse of a function and graph both functions on the same axes.
Inverses
General Definition:
Inverses represent the concept of "undoing" an action.
Everyday Examples:
Inverses can be thought of in simple terms:
What is the inverse of putting on your shoes? (Taking them off)
What is the inverse of opening a door? (Closing it)
Some operations are inherently easier to undo than others:
For instance, tying shoes may not have a straightforward process compared to untying.
Mathematical Inverses:
For addition, the inverse operation is subtraction.
For multiplication, the inverse is division.
Definition of Inverse Relation
To define the inverse of a relation:
Identify the specific input for a given output by exchanging inputs (x) and outputs (y).
For a relation R, the inverse relation is denoted as R⁻¹, defined as:
R⁻¹ = {(y, x) | (x, y) ∈ R}
Important Notes:
Domain and range of the original relation are exchanged in the inverse.
This means that the outputs of the original function become the inputs of the inverse function and vice versa.
Exercises
Exercise 1:
Determine the inverse of the relation R = {(5, 7), (6, 2), (6, 3), (2, −1)}.
Graph R and its inverse, and determine the domain and range for both.
Exercise 2:
Determine the inverse of the relation y = -|x| + 2.
Find four points of the inverse and determine its domain and range.
One-to-One Functions
A function is one-to-one if every input corresponds to one unique output.
Characteristics:
If f(x₁) = f(x₂) implies x₁ = x₂, then f is one-to-one.
Visual Test:
One-to-one functions pass the horizontal line test (no horizontal line intersects the graph at more than one point).
Inverse Functions and Their Properties
If f is one-to-one:
Each output has exactly one input.
The function and its inverse (f⁻¹) will also be functions.
If f passes the horizontal line test, then f⁻¹ will pass the vertical line test, confirming that the inverse is also a function.
Exercises and Tasks
Exercise 3:
Determine if the function f(x) = (x + 3)³ has an inverse.
Finding Inverse Function Formulas:
To find inverses of functions defined by formulas:
Replace f(x) with y.
Switch x and y in the equation.
Solve for y.
Rename y as f⁻¹(x).
Exercise 4:
Find the inverse of the function f(x) = 6 - 1/5.
Properties of Inverses
Fundamental Property:
Doing an operation and then its inverse should yield the original value:
f(f⁻¹(x)) = x for all x in the domain of f⁻¹.
f⁻¹(f(x)) = x for all x in the domain of f.
Important Note:
These properties do not hold if the function is not one-to-one.
Example: Verifying the inverses with f(x) = √(y²) leads to y ≠ -y unless y is non-negative.
Exercises to Verify Relationships
Exercise 5:
Verify the relationships for given functions:
(f⁻¹(f(x)) = x)
(f(f⁻¹(x)) = x)
Graphical Representation of Inverses
Observations:
The graph of an inverse function is a reflection of the original function across the line y = x.
This reflection property helps to visually confirm that the two functions are indeed inverses of one another.
Example:
For the function f(x) = (x - 1)³ + 2, its inverse can be determined and graphed accordingly, demonstrating how the output of f maps to the input of its inverse and vice versa.