The Main Idea 

• Definition of Inverse Functions:
  • An inverse function “undoes” what the original function does
  • Notation: [latex]f^{-1}[/latex] is the inverse of [latex]f[/latex]
  • [latex]f^{-1}(f(x)) = x[/latex] and [latex]f(f^{-1}(x)) = x[/latex]
• Domain and Range Relationship:
  • Domain of [latex]f[/latex] becomes the range of [latex]f^{-1}[/latex]
  • Range of [latex]f[/latex] becomes the domain of [latex]f^{-1}[/latex]
• One-to-One Functions:
  • Each element in the codomain is paired with at most one element in the domain
  • Only one-to-one functions have inverses that are also functions
• Horizontal Line Test:
  • Determines if a function is one-to-one
  • A function passes if no horizontal line intersects its graph more than once
• Inverse Function Properties:
  • Reverses the input and output of the original function
  • Graphically, it’s a reflection of the original function over [latex]y = x[/latex]

The Main Idea 

The process for finding an inverse function is:

1. Ensure the function is one-to-one
2. Replace [latex]f(x)[/latex] with [latex]y[/latex]
3. Solve the equation for [latex]x[/latex]
4. Interchange [latex]x[/latex] and [latex]y[/latex]
5. Replace [latex]y[/latex] with [latex]f^{-1}(x)[/latex]

You can always check and verify if you have found a functions inverse by:

• Checking that [latex]f^{-1}(f(x)) = x[/latex]
• Also verifying that [latex]f(f^{-1}(x)) = x[/latex]

The Main Idea 

• Graphical Relationship of Inverses:
  • Inverse functions are reflections of each other over the line [latex]y = x[/latex]
  • Points [latex](a,b)[/latex] on [latex]f(x)[/latex] correspond to points [latex](b,a)[/latex] on [latex]f^{-1}(x)[/latex]
• Graphing Inverse Functions:
  • Plot the original function
  • Reflect points over [latex]y = x[/latex]
  • Connect reflected points to form the inverse function’s graph
• Restricted Domains:
  • Used when a function is not one-to-one over its entire domain
  • Allows creation of an inverse function for a portion of the original function
  • Different restrictions can lead to different inverse functions
• One-to-One on Restricted Domains:
  • Ensure the function passes the horizontal line test on the restricted domain
  • Domain of [latex]f[/latex] becomes range of [latex]f^{-1}[/latex]and vice versa

The Main Idea 

• Definition of Inverse Trigonometric Functions:
  • Created by restricting domains of standard trig functions
  • Inverse sine ([latex]\sin^{-1}[/latex] or arcsin), inverse cosine ([latex]\cos^{-1}[/latex] or arccos), inverse tangent ([latex]\tan^{-1}[/latex] or arctan), etc.
• Domains and Ranges:
  • arcsin and arccos: Domain [latex][-1, 1],[/latex] Range [latex][\frac{-π}{2}, \frac{π}{2}][/latex] (arcsin) and [latex][0, π][/latex] (arccos)
  • arctan: Domain [latex](-\infty, \infty)[/latex], Range [latex][\frac{-π}{2}, \frac{π}{2}][/latex]
• Graphs:
  • Reflections of restricted trigonometric functions over [latex]y = x[/latex]
• Composition Properties:
  • [latex]\sin(\sin^{-1}(x)) = x[/latex] for [latex]x[/latex] in [latex][-1, 1][/latex]
  • [latex]\sin^{-1}(\sin(x)) = x[/latex] for [latex]x[/latex] in [latex][\frac{-π}{2}, \frac{π}{2}][/latex]
  • Similar properties for other inverse trig functions