Study Notes on Inverse Functions

Inverse Functions

Definition of Inverse Function

  • A function $f$ has an inverse function $g$, denoted as $f^{-1}$, if for each element $x$ in the domain of $f$, there is a unique element $y$ in the domain of $g$ such that:
    f(g(y))=yf(g(y)) = y
    and g(f(x))=x.g(f(x)) = x.

Conditions for Inverses

  • For a function to possess an inverse, it must be bijective, meaning:
    1. Injective (One-to-One): No two different inputs map to the same output.
    2. Surjective (Onto): Every element in the co-domain is an output of the function.

Example Function: $f(x)$

  • Let the function $f$ be defined by:
    f: ext{PC}
    ightarrow ext{Inac} ext{ where } ac < 0
  • This notation indicates the function operates over a specific set of numbers where $ac$, the product of coefficients, is less than zero.

Sketching the Graph of $f(x)$

  • To demonstrate that $f(x)$ has an inverse, we need to sketch the graph of $f(x)$. It is essential to:
    1. Plot the points for selected inputs to establish the shape of the function.
    2. Ensure the graph clearly shows that it passes the horizontal line test (indicating it is one-to-one).
  • A typical function might be a concave graph in the context of restrictions posed on $ac$.

Graph Reflection for Inverse Functions

  • An important property of inverse functions is that the graph of $y = f(x)$ is a reflection of the graph of $y = g(x)$ about the line $y = x$.
  • This means that:
    • If the original function $f(x)$ is graphed in the Cartesian plane, drawing the line $y = x$ and reflecting points across this line will yield the graph representing the inverse function $g(x)$.

Implications

  • When sketching both functions on the same graph:
    • Use dashed lines for the line $y = x$ to indicate the reflection line.
    • Highlight the intersections where $f(x) = g(x)$, which occurs when $x$ is the same for both functions.

Summary of Key Points

  • The existence of an inverse function requires the original function to be bijective.
  • Inverse functions are graphically represented by reflecting across the line $y = x$.
  • To prove that a function has an inverse, one can demonstrate bijectivity and explore the graphical representation.