Function Composition, Decompositions, and Geometric Transformations

Fundamentals of Function Composition

• Definition of Composite Function:

  • A composite function applies one function to the result of another function.

  • The output of the inner function becomes the direct input of the outer function.

  • Conceptual flow of values: $\text{Input } x \longrightarrow \text{Inner Function } g(x) \longrightarrow \text{Output } g(x) \longrightarrow \text{Outer Function } f(g(x)) \longrightarrow \text{Final Output}$.

• Notation and Terminology:

  • Written algebraically as $(f \circ g)(x)$ or $f(g(x))$.

  • Read aloud as "$f$ of $g$ of $x$".

  • In the expression $f(g(x))$, $g(x)$ is the inner function and $f(x)$ is the outer function.

  • In the expression $g(f(x))$, $f(x)$ is the inner function and $g(x)$ is the outer function.

Algebraic Function Composition

• Basic Algebraic Composition (Example 1):

  • Given the base functions:

  • $f(x) = x^2 + 3$

  • $g(x) = 2x - 1$

  • Finding $(f \circ g)(x)$ / $f(g(x))$:

  • Substitute the entire expression for $g(x)$ into every instance of $x$ within $f(x)$:

    • $f(g(x)) = f(2x - 1)$

  • Apply the rule for $f$:

    • $f(2x - 1) = (2x - 1)^2 + 3$

  • Expand the binomial squared:

    • $(2x - 1)^2 = (2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1$

  • Combine like terms:

    • $4x^2 - 4x + 1 + 3 = 4x^2 - 4x + 4$

  • Final simplified expression:

    • $(f \circ g)(x) = 4x^2 - 4x + 4$

  • Finding $(g \circ f)(x)$ / $g(f(x))$:

  • Substitute the entire expression for $f(x)$ into every instance of $x$ within $g(x)$:

    • $g(f(x)) = g(x^2 + 3)$

  • Apply the rule for $g$:

    • $g(x^2 + 3) = 2(x^2 + 3) - 1$

  • Distribute the constant $2$:

    • $2(x^2) + 2(3) - 1 = 2x^2 + 6 - 1$

  • Combine constants:

    • $2x^2 + 5$

  • Final simplified expression:

    • $(g \circ f)(x) = 2x^2 + 5$

  • Core Principle of Non-Commutativity:

  • Function composition is generally not commutative:

    • $(f \circ g)(x) \neq (g \circ f)(x)$

• $4x^2 - 4x + 4 \neq 2x^2 + 5$

Numerical Evaluation of Composite Functions

• Evaluating at a Specific Input Value (Example 2):

  • Given $f(x) = x^2 + 3$ and $g(x) = 2x - 1$, evaluate $(f \circ g)(4)$.

  • Method: Inside-Out Stepwise Evaluation:

  • Step 1: Evaluate the inner function $g(x)$ at $x = 4$:

    • $g(4) = 2(4) - 1$

    • $g(4) = 8 - 1 = 7$

  • Step 2: Use the resulting output ($7$) as the input for the outer function $f(x)$:

    • $f(7) = (7)^2 + 3$

    • $f(7) = 49 + 3 = 52$

  • Final value:

  • $(f \circ g)(4) = 52$

Composition with Domain Restrictions

• Determining Composite Formulas and Restricted Domains (Example 3):

  • Given the functions:

  • $f(x) = \frac{1}{x}$

  • $g(x) = \sqrt{x - 5}$

  • Finding the Composite Function $(f \circ g)(x)$:

  • Substitute $g(x)$ into $f(x)$:

    • $(f \circ g)(x) = f(\sqrt{x - 5}) = \frac{1}{\sqrt{x - 5}}$

  • Determining the Domain of $f(g(x))$:

  • The domain of a composite function $f(g(x))$ requires that two conditions are simultaneously satisfied:

    • Condition 1 (Inner Function Domain): The input $x$ must reside within the domain of the inner function $g(x)$.

    • For $g(x) = \sqrt{x - 5}$, the radicand must be non-negative:

      • $x - 5 \ge 0$

      • $x \ge 5$

    • Condition 2 (Outer Function Domain): The output of $g(x)$ must reside within the domain of the outer function $f(x)$.

    • For $f(u) = \frac{1}{u}$, the denominator cannot equal zero:

      • $u \neq 0 \implies g(x) \neq 0$

• $\sqrt{x - 5} \neq 0$

• $x - 5 \neq 0$

• $x \neq 5$

• Synthesizing the Restrictions:

  • Combining $x \ge 5$ and $x \neq 5$ eliminates the endpoint $5$:

• Inequality notation: $x > 5$

• Interval notation: $(5, \infty)$

Absolute Value Transformations: $|f(x)|$ vs. $f(|x|)$

• Overview of Absolute Value Composition:

  • Composing any base function $f(x)$ with the absolute value function $g(x) = |x|$ produces distinct geometric transformations depending entirely on whether the composition is outer or inner.

• Outer Composition: $y = |f(x)|$ (or $g(f(x))$ where $g(x) = |x|$):

  • Conceptual Meaning:

  • The absolute value acts on the outside of the function, directly modifying the output values ($y$-outputs): $|y|$.

  • Transformation Rules:

  • Any portion of the graph of $f(x)$ that lies below the $x$-axis ($y < 0$) is reflected across the $x$-axis into the positive half-plane ($y > 0$).

  • Any portion of the graph of $f(x)$ that lies on or above the $x$-axis ($y \ge 0$) remains completely unchanged.

• Inner Composition: $y = f(|x|)$ (or $f(g(x))$ where $g(x) = |x|$):

  • Conceptual Meaning:

  • The absolute value acts on the inside of the function, directly modifying the input values ($x$-inputs) before $f$ processes them.

  • Transformation Rules:

  • Delete and discard the entire left side of the graph where $x < 0$.

  • Retain the right side of the graph where $x \ge 0$.

  • Reflect the retained right side ($x \ge 0$) across the $y$-axis onto the left side ($x < 0$).

  • Operational memory aids: "THINK SYMMETRY" and "KEEP RIGHT SIDE SAME".

  • The resulting graph exhibits even symmetry (line symmetry across the $y$-axis).

Graphical Transformation Analysis and Practice

• Analysis of Example 4 Base Function $y = f(x)$ on Domain $[-4, 4]$:

  • Piecewise linear graph featuring the following key anchor coordinates:

  • Endpoint at $(-4, -3)$

  • Line segment running from $(-4, -3)$ to $(-1, -1)$, with an $x$-intercept at $(-1.75, 0)$

  • Horizontal segment running from $(-1, -1)$ to $(0, -1)$

  • Line segment from $(0, -1)$ rising through an $x$-intercept at $(1.25, 0)$ to a local maximum peak at $(1.75, 0.75)$

  • Falling segment from $(1.75, 0.75)$ down to the right endpoint at $(4, 0)$

• Part A: Constructing $y = g(f(x)) = |f(x)|$:

  • Modifies all $y$-values by taking their magnitude: $y \rightarrow |y|$.

  • Transformation applied point-by-point:

  • Left endpoint: $(-4, -3) \longrightarrow (-4, 3)$

  • Intercept remains: $(-1.75, 0) \longrightarrow (-1.75, 0)$

  • Corner point: $(-1, -1) \longrightarrow (-1, 1)$

  • Segment along $y = -1$ flips to $y = 1$: horizontal line segment from $(-1, 1)$ to $(0, 1)$

  • Segment below the axis from $(0, -1)$ to $(1.25, 0)$ flips to run from $(0, 1)$ down to $(1.25, 0)$

  • Positive region stays unchanged: passes through peak $(1.75, 0.75)$ and terminates at $(4, 0)$

• Part B: Constructing $y = f(g(x)) = f(|x|)$:

  • Preserves the right side ($x \ge 0$) and reflects it across the $y$-axis:

  • The portion for $x < 0$ is completely deleted.

  • Retained right side ($x \ge 0$):

    • Center vertex at $(0, -1)$

    • Rises through $x$-intercept $(1.25, 0)$

    • Reaches peak at $(1.75, 0.75)$

    • Drops to endpoint at $(4, 0)$

  • Reflected left side ($x < 0$):

    • Starts at center vertex $(0, -1)$

    • Rises through symmetric $x$-intercept at $(-1.25, 0)$

    • Reaches symmetric peak at $(-1.75, 0.75)$

    • Drops to symmetric endpoint at $(-4, 0)$

Decomposing Functions

• Principles of Function Decomposition:

  • Decomposing a function involves reversing composition to identify the constituent inner function and outer function.

  • Standard target: Express a combined function as $h(x) = g(f(x))$ or $h(x) = f(g(x))$.

  • Common Structural Locations for the Inner Function:

  • Grouped inside parentheses: $(expression)^n$

  • Positioned underneath a radical symbol: $\sqrt{expression}$

  • Located in the denominator of a fraction: $\frac{1}{expression}$

• Identification Practice (Example 5): Expressing $h(x) = g(f(x))$:

  • Note: In this specific problem configuration, $f(x)$ is designated as the inner function and $g(x)$ is designated as the outer function.

  • Problem 1: $h(x) = (3x - 5)^5$

  • Inner function (inside the parentheses): $f(x) = 3x - 5$

  • Outer function (power of 5 rule): $g(x) = x^5$

  • Verification: $g(f(x)) = g(3x - 5) = (3x - 5)^5$

  • Problem 2: $h(x) = \frac{1}{x^2 + 3}$

  • Inner function (in the denominator): $f(x) = x^2 + 3$

  • Outer function (reciprocal rule): $g(x) = \frac{1}{x}$

  • Verification: $g(f(x)) = g(x^2 + 3) = \frac{1}{x^2 + 3}$

  • Problem 3: $h(x) = \sqrt{2x + 1}$

  • Inner function (underneath the square root radical): $f(x) = 2x + 1$

  • Outer function (square root rule): $g(x) = \sqrt{x}$

  • Verification: $g(f(x)) = g(2x + 1) = \sqrt{2x + 1}$

Real-World Applications of Function Composition

• Search and Rescue Modeling Scenario (Example 6):

  • Context: A search and rescue team estimates that a lost hiker travels outward from a search site at an average rate of $2\,\text{mph}$.

  • Given functional models:

  • Radius function: $g(x) = 2x$, representing the radius of the search zone (in miles) after $x$ hours.

  • Area function: $f(r) = \pi r^2$, representing the total area of a circular search zone (in square miles) given a radius $r$.

• Part 1: Deriving the Composite Function:

  • Construct $(f \circ g)(x)$ to represent total search area directly as a function of elapsed time $x$:

  • Substitute the radius function $g(x)$ into the area function $f(r)$:

    • $f(g(x)) = \pi (g(x))^2$

    • $f(g(x)) = \pi (2x)^2$

  • Square the inner term:

    • $(2x)^2 = 4x^2$

  • Multiply by $\pi$:

    • $f(g(x)) = 4\pi x^2$

  • Final composite model:

  • $(f \circ g)(x) = 4\pi x^2$

• Part 2: Calculating Theoretical Area for Given Time:

  • Scenario: Calculate the theoretical search area if the hiker has been missing for $6$ hours ($x = 6$):

  • Substitute $x = 6$ into the composite formula:

    • $f(g(6)) = 4\pi (6)^2$

    • $f(g(6)) = 4\pi (36)$

    • $4 \times 36 = 144$

    • $f(g(6)) = 144\pi$

  • Theoretical search area:

  • Exact area: $144\pi\,\text{square miles}$

• Part 3: Contextual Meaning of Variables:

  • Input ($x$): Represents the time elapsed in hours since the hiker went missing.

  • Output ($f(g(x))$): Represents the total search area in square miles covered by the circular search zone.