Function transformations and inverses Knowledge-Based Study

Study notes

1. Rigid Transformations: Translations

   A transformation alters the position, orientation, or shape of a parent function $f(x)$. The simplest class of transformations is rigid transformations, which shift the graph without altering its basic shape or size.

   Vertical Translations

   Given the function $y = f(x)$ and a positive real constant $k > 0$:

   • $y = f(x) + k$ shifts the graph of $f(x)$ upward by $k$ units. Every point $(x, y)$ maps to $(x, y + k)$.
   • $y = f(x) - k$ shifts the graph of $f(x)$ downward by $k$ units. Every point $(x, y)$ maps to $(x, y - k)$.

   Horizontal Translations

   When an adjustment is made directly to the input variable $x$ prior to applying the function rule, a horizontal shift occurs. For $h > 0$:

   • $y = f(x - h)$ shifts the graph to the right by $h$ units. The point $(x, y)$ maps to $(x + h, y)$.
   • $y = f(x + h)$ shifts the graph to the left by $h$ units. The point $(x, y)$ maps to $(x - h, y)$.

   Key Principle: Operations applied outside the function rule affect the output ($y$-values) directly, whereas operations applied inside the function argument produce an inverse effect on the input ($x$-values).

   Non-Rigid Transformations and Reflections

   Transformations that stretch, compress, or reflect the graph change its orientation or rate of growth.

   Reflections Across Coordinate Axes

   • Vertical Reflection: The function $y = -f(x)$ reflects the graph across the $x$-axis. Every point $(x, y)$ maps to $(x, -y)$.
   • Horizontal Reflection: The function $y = f(-x)$ reflects the graph across the $y$-axis. Every point $(x, y)$ maps to $(-x, y)$.

   Stretches and Compressions (Dilations)

   Multiplying by a constant scale factor scales the graph either vertically or horizontally:

   • Vertical Scaling ($y = a f(x)$): If $|a| > 1$, the graph is stretched vertically by a factor of $|a|$. If $0 < |a| < 1$, it is compressed vertically by a factor of $|a|$.
   • Horizontal Scaling ($y = f(b x)$): If $|b| > 1$, the graph is compressed horizontally by a factor of $\frac{1}{|b|}$. If $0 < |b| < 1$, it is stretched horizontally by a factor of $\frac{1}{|b|}$.

   Order of Combined Transformations

   When applying multiple transformations of the form $g(x) = a f(b(x - h)) + k$, follow a consistent order:

   1. Horizontal transformations: factor inside as $b(x - h)$ to identify the horizontal scale factor $\frac{1}{b}$ and the horizontal shift $h$.
   2. Reflections and scalings: apply horizontal and vertical scalings and reflections.
   3. Vertical shift: apply the vertical translation by $k$ units last.

   Understanding Inverse Functions

   An inverse function reverses the mapping of the original function. If a function $f$ maps an element $x$ in its domain to $y$ in its range, then its inverse $f^{-1}$ maps $y$ back to $x$.

   The One-to-One Criterion

   A function $f$ possesses an inverse function $f^{-1}$ if and only if it is bijective (one-to-one and onto its range):

   • Algebraic Definition: A function is one-to-one (injective) if $f(x_1) = f(x_2)$ implies $x_1 = x_2$.
   • Geometric Criterion (Horizontal Line Test): A function has an inverse on a given interval if every horizontal line intersects its graph at most once.

   Domain and Range Interplay

   The operational roles of domain and range interchange completely between a function and its inverse:

   • $\text{Domain}(f^{-1}) = \text{Range}(f)$
   • $\text{Range}(f^{-1}) = \text{Domain}(f)$

   Geometric Relationship

   The graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ across the line $y = x$. Consequently, if the point $(a, b)$ lies on the graph of $f$, then $(b, a)$ must lie on the graph of $f^{-1}$.

   Algebraic Procedure for Finding Inverses

   To find the inverse formula for a one-to-one function $y = f(x)$:

   1. Verify that the function is one-to-one on its domain (or restrict the domain where necessary).
   2. Swap the variables $x$ and $y$ to reflect the interchange of input and output: $x = f(y)$.
   3. Solve the resulting equation explicitly for $y$.
   4. Express the solved expression as $f^{-1}(x)$.

   Example: Linear and Rational Inverses

   Consider the rational function $f(x) = \frac{2x + 1}{x - 3}$ for $x \neq 3$:

   • Set $x = \frac{2y + 1}{y - 3}$.
   • Multiply both sides by $y - 3$: $x(y - 3) = 2y + 1$.
   • Collect terms containing $y$: $xy - 2y = 3x + 1$.
   • Factor out $y$: $y(x - 2) = 3x + 1$.
   • Isolate $y$: $f^{-1}(x) = \frac{3x + 1}{x - 2}$, where $x \neq 2$.

   The Composition Identity

   A fundamental property used to verify that two functions $f$ and $g$ are inverses of one another is the composition identity:

   $$(f \circ g)(x) = f(g(x)) = x \quad \text{for all } x \in \text{Domain}(g)$$$$(g \circ f)(x) = g(f(x)) = x \quad \text{for all } x \in \text{Domain}(f)$$