Composite functions and inverse functions are powerful tools in algebra. They allow us to combine existing functions in new ways and reverse their effects. These concepts are crucial for understanding more complex mathematical relationships and solving real-world problems.

Function notation and transformations help us describe and manipulate functions visually and algebraically. By mastering these skills, we can analyze and modify functions more effectively, opening up new possibilities in mathematical modeling and problem-solving.

Composite Functions

Construction of composite functions

• Formed by combining two or more functions into a single function (function composition)
• Denoted as $(f \circ g)(x)$ or $f(g(x))$ where $g(x)$ is the inner function and $f(x)$ is the outer function
• Evaluated by first applying the inner function $g(x)$ to the input value $x$, then applying the outer function $f(x)$ to the result of $g(x)$
  • Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = f(g(x)) = (x + 1)^2$
• Algebraic method constructs composite functions by substituting the inner function $g(x)$ into the outer function $f(x)$ wherever $x$ appears, then simplifying the resulting expression
  • Example: If $f(x) = 2x - 1$ and $g(x) = 3x + 2$, then $(f \circ g)(x) = f(g(x)) = 2(3x + 2) - 1 = 6x + 3$
• Graphical method constructs composite functions by tracing the input value $x$ through the graphs of $g(x)$ and $f(x)$
  • Locate $x$ on the $x$-axis, move vertically to the graph of $g(x)$, then horizontally to the $y$-axis to find $g(x)$
  • Treat $g(x)$ as the input for $f(x)$, move vertically to the graph of $f(x)$, then horizontally to the $y$-axis to find $(f \circ g)(x)$

Inverse Functions

Identification of one-to-one functions

• One-to-one functions map each input value to a unique output value
  • Also known as injective functions
• Passes the horizontal line test where any horizontal line intersects the graph of the function at most once
  • Example: $f(x) = x^3$ is one-to-one, but $f(x) = x^2$ is not
• Only one-to-one functions have inverses because each output value corresponds to a unique input value
  • Non-one-to-one functions cannot have inverse functions due to multiple inputs mapping to the same output
• A function that is both one-to-one and onto (surjective) is called a bijective function and always has an inverse

Determination of inverse functions

• Inverse functions "undo" the operation of the original function (inverse operation)

• Denoted as $f^{-1}(x)$ and satisfies the property: if $f(a) = b$, then $f^{-1}(b) = a$

  • Example: If $f(x) = 2x + 1$, then $f^{-1}(x) = \frac{x - 1}{2}$

• Algebraic method to find inverse functions:

  1. Replace $f(x)$ with $y$
  2. Swap $x$ and $y$ variables
  3. Solve the equation for $y$
  4. Replace $y$ with $f^{-1}(x)$

• Graphical method to find inverse functions:

  • Reflect the graph of the original function across the line $y = x$
  • The domain of $f(x)$ becomes the range of $f^{-1}(x)$ and vice versa

• Domain and range restrictions:

  • The domain of the inverse function is the range of the original function
  • The range of the inverse function is the domain of the original function
  • Any restrictions on the domain of the original function become restrictions on the range of the inverse function
    • Example: If $f(x) = \sqrt{x}$, then $f^{-1}(x) = x^2$, but the domain of $f^{-1}(x)$ is restricted to non-negative real numbers

Function Notation and Transformations

• Function notation is a way of writing functions using symbols, typically in the form f(x) = expression
• Function transformations involve changing the graph of a function through operations such as translations, reflections, and scaling
• These transformations can be represented using function notation, allowing for clear communication of how a function's graph is modified