Notes on Inverse Functions

Section 4.1: Inverse Functions

Composition of Functions

  • Recall the concept of function composition.

  • For functions $f$ and $g$, the composition is defined as:

    • $g(f(x))$ means substituting $f(x)$ into $g$.

  • Example provided in the transcript:

    • Let $f(x) = x^2$ and $g(x) = rac{3}{4}x + 4$.

Definition of Inverse Functions

  • Functions $f$ and $g$ are considered inverse functions if:

    1. The composition $f(g(x)) = x$ for all $x$ in the domain of $g$.

    2. The composition $g(f(x)) = x$ for all $x$ in the domain of $f$.

  • This means that they effectively undo each other.

Finding the Inverse Function

  • Process to derive the inverse for a given function involves:

    • Step 1: Replace $f(x)$ with $y$.

    • Step 2: Solve for $x$ in terms of $y$.

    • Step 3: Swap $x$ and $y$ in the resulting equation to find $f^{-1}(x)$.

  • Example Process:

    • If $f(x) = x^2$, to find $f^{-1}(x)$:

    • Set $y = x^2$.

    • Solve for $x$: $x = ext{sqrt}(y)$ (or $x = ext{-sqrt}(y)$ depending on context).

    • Taking the square root gives inverse behavior of squaring.

Important Facts About Inverses

  1. If $f(a) = b$, then $f^{-1}(b) = a$. This shows the reversibility property.

  2. The domain of the original function $f$ corresponds to the range of the inverse function $f^{-1}$;

  3. The range of the original function $f$ corresponds to the domain of the inverse function $f^{-1}$.

Graphing the Inverse Function

  • To find $f^{-1}(x)$ graphically, sketch the graph of $y = f(x)$, then reflect it across the line $y = x$.

  • Example: Graph of $y = f(x) = x^2$, the inverse would be a reflection.

One-to-One Function Requirement

  • A function $f$ must be a one-to-one function to have an inverse.

  • Horizontal Line Test: If a horizontal line intersects the graph of the function at more than one point, then the function does not have an inverse.

Algebraic Process to Find the Inverse Function

Steps:

  1. Set $y = f(x)$.

  2. Interchange the roles of $x$ and $y$.

  3. Solve for $y$ to find $f^{-1}(x)$.

Example Problem:

Find $f^{-1}(x)$ for the function $f(x) = x^3 + 3$:

  • Step 1: Set $y = x^3 + 3$.

  • Step 2: Interchange:

    • $x = y^3 + 3$.

  • Step 3: Solve for $y$:

    • $y^3 = x - 3$ → $y = oxed{ ext{cube root}(x - 3)}$.

    • Thus, $f^{-1}(x) = ext{cube root}(x - 3)$.

  • Example outcome shows that $f^{-1}(x)$ reverts $f(x)$ back to $x$.

Failure of Invertibility

  • If a function fails the horizontal line test, it cannot be inverted:

    • Example: The function $f(x) = x^2$ restricted to $x
      eq 0$.

Constraints on Inverse Functions

  • Additional requirement: The function must also fulfill conditions regarding the values of $x$ to maintain physical and mathematical relevance (e.g., square roots).