Inverse Functions Explained

• An inverse function reverses the action of another function
• If f(x) produces y, then the inverse function, denoted as f⁻¹(x), produces x from y
• f⁻¹(y) = x

Notation

• Given a function f(x), its inverse is denoted as f⁻¹(x)
• f⁻¹(x) is read as "f inverse of x"
• The "-1" is not an exponent; it indicates an inverse function

Finding the Inverse Function

• Replace f(x) with y
• Swap x and y in the equation
• Solve the equation for y
• Replace y with f⁻¹(x)

Example: Finding the Inverse

• Given f(x) = 2x + 3:
  • Replace f(x) with y: y = 2x + 3
  • Swap x and y: x = 2y + 3
  • Solve for y:
    • x - 3 = 2y
    • y = (x - 3) / 2
  • Replace y with f⁻¹(x): f⁻¹(x) = (x - 3) / 2

Domain and Range

• The domain of f(x) becomes the range of f⁻¹(x)
• The range of f(x) becomes the domain of f⁻¹(x)
• Domain and range are swapped

One-to-One Functions

• A function must be one-to-one to have an inverse function
• A one-to-one function passes both the vertical and horizontal line tests
• The Vertical Line Test checks if a graph is a function; a vertical line intersects the graph at most once
• The Horizontal Line Test determines if a function has an inverse; a horizontal line intersects the graph at most once

Composition of Inverse Functions

• If f(x) and g(x) are inverse functions, then:
  • f(g(x)) = x
  • g(f(x)) = x
• This property can be used to verify if two functions are inverses of each other
• The composition results in the original input x

Example: Composition to Verify Inverses

• Given f(x) = 2x + 3 and g(x) = (x - 3) / 2
  • f(g(x)) = 2((x - 3) / 2) + 3 = (x - 3) + 3 = x
  • g(f(x)) = ((2x + 3) - 3) / 2 = (2x) / 2 = x
• Since both compositions result in x, f(x) and g(x) are inverse functions

Graphing Inverse Functions

• The graph of f⁻¹(x) is the reflection of the graph of f(x) over the line y = x
• To graph an inverse, reflect each point (a, b) on f(x) to (b, a) on f⁻¹(x)

Steps for Finding and Graphing Inverses

• Given f(x), find f⁻¹(x) algebraically
• Create a table of values for f(x)
• Swap the x and y values to create a table for f⁻¹(x)
• Plot both sets of points and draw the curves
• The graphs should be reflections of each other across y = x

Restricted Domains

• If a function is not one-to-one over its entire domain, it may be possible to restrict the domain to create a one-to-one function
• For example, f(x) = x² does not pass the horizontal line test
• By restricting the domain to x ≥ 0, the new function is one-to-one and has an inverse

Inverse of Quadratic Function with Restricted Domain

• Given f(x) = x² for x ≥ 0
  • Replace f(x) with y: y = x²
  • Swap x and y: x = y²
  • Solve for y: y = √x (since x ≥ 0, y ≥ 0)
  • f⁻¹(x) = √x

Key Properties Recap

• Notation uses a superscript -1: f⁻¹(x)
• Swap x and y to find the inverse: Algebraically swapping of x and y and solving for y
• One-to-one functions are required: Passes both vertical and horizontal line tests
• Compositions equal x: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
• Reflection over y = x: Geometrically, graphs reflected over the line y = x
• Domain and range swap: The domain of f(x) is the range of f⁻¹(x), and vice versa