Introduction to Composite and Inverse Functions

Course Administration and Exam Logistics

Mid-Semester Exam Feedback and Marking: * Exams have been received by the mathematics office. * Submissions will be scanned into Gradescope for marking. * The marking process is estimated to take approximately two weeks. * Results will be returned to students via the same method as assignments. * Performance on the in-semester exam does not determine failure of the entire course; students can still pass the hurdle requirement via the final exam.
Deferred Exams: * The deferred exam is scheduled for the Saturday after next, not the upcoming Saturday. * The specific timetable for the deferred exam is expected to be released early next week; students should monitor their email for updates. * Marked in-semester exams and the official exam paper cannot be released until the deferred exam has been completed. * Part A marks may be released earlier than Part B as they are quicker to evaluate.
Upcoming Assessments: * Week 10 Collaborative Task: An assessed collaborative activity regarding functions is being held in classes this week. * Final Assignment: The second and final assignment was released recently and is due not this Friday, but the following Friday. * Applied Classes: Students will have dedicated time to work on the final assignment during next week’s applied classes.
Final Exam Scope: * The final exam focuses on material not covered in the in-semester exam. * There will be no explicit questions dedicated solely to vectors, applications of vectors, or complex numbers. * However, students must maintain their vector knowledge, as Chapter 6 (Lines and Planes) is heavily dependent on vector operations.

Overview of Composition of Functions

Review of Inequalities (Section 7.3): * A critical mathematical rule for solving inequalities is that whenever both sides are multiplied or divided by a negative quantity, the inequality symbol must reverse direction.
Introduction to Section 7.4: Composition of Functions: * Composition is often referred to as a "function of a function." * It involves the output of one function becoming the input for another. * Real-World Metaphors for Order of Operations: * Car Assembly Line: In a factory, the carpet is typically installed before the seats. If the order is reversed (seats before carpet), the carpet would have to be cut to fit, increasing labor and time complexity. This illustrates that in many systems, order matters. * Baking: Recipes usually require mixing dry ingredients (flour, sugar, baking powder) before adding wet ingredients (eggs, milk). Mixing wet ingredients with a single dry component like baking powder first can result in a "wonky" cake.
Composition where Order is Inconsequential (Anomalies): * The example of holiday pay loading demonstrates a scenario where the order of functions does not change the result. * Scenario: Working a 4-hour shift at a standard rate of $20.00 per hour with a double-time ($2 imes$) loading for a public holiday. * Method 1 (Total then Load): Calculate total pay first $(4 imes 20 = 80)$ and then apply the bonus loading $(80 imes 2 = 160)$ . * Method 2 (Rate then Total): Calculate the holiday hourly rate first $(20 imes 2 = 40)$ and then multiply by hours worked $(40 imes 4 = 160)$ . * In mathematics, however, this commutative property is an anomaly; for most composite functions, order is vital.

Composite Function Notation and Evaluation

Notation Types: 1. Function Representation: $I(h(x))$ 2. Circle Notation: $(f ext{ o } g)(x)$ , often found in North American textbooks, which is equivalent to $f(g(x))$ .
Evaluation Principles: * Always begin evaluating from the innermost function and work outward, following standard order of operations. * The inner function must be defined for its output to serve as a valid input for the external function.
Worked Examples with Numerical Inputs: * Defined Functions: * $f(x) = x + 1$ * $h(x) = 4x$ * $I(x) = x^2$ * $j(x) = \sqrt{x}$ * Evaluating $f(h(-2))$ : 1. Inner function: $h(-2) = 4 \times (-2) = -8$ . 2. Outer function: $f(-8) = -8 + 1 = -7$ . * Evaluating $j(f(8))$ : 1. Inner function: $f(8) = 8 + 1 = 9$ . 2. Outer function: $j(9) = \sqrt{9} = 3$ . (Note: We use the principal/positive root to maintain the unique output definition of a function).
Worked Examples with Symbolic Inputs: * Evaluating $I(h(x))$ : 1. Replace $h(x)$ with its definition: $I(4x)$ . 2. Apply functionality of $I$ : $(4x)^2 = 16x^2$ . * Evaluating $h(I(x))$ : 1. Replace $I(x)$ with its definition: $h(x^2)$ . 2. Apply functionality of $h$ : $4(x^2) = 4x^2$ . * Comparison: $16x^2 \neq 4x^2$ , proving that order of composition typically changes the resulting function.
Algebraic Solution for Equating Composite Functions: * To find where $16x^2 = 4x^2$ : 1. $16x^2 - 4x^2 = 0$ 2. $12x^2 = 0$ 3. $x = 0$ * Note: One should not divide by $x^2$ as it rules out the possibility of $x=0$ and is mathematically invalid if $x=0$ .

Domain and Range of Composite Functions

Case Study Example: * $f(x) = \sqrt{x - 5}$ * $g(x) = x^2 - 4$
Analyzing $f(g(x))$ : * Expression: $f(x^2 - 4) = \sqrt{(x^2 - 4) - 5} = \sqrt{x^2 - 9}$ . * Domain Analysis: The term under a square root must satisfy the inequality $x^2 - 9 \geq 0$ . * Solving the inequality $x^2 \geq 9$ : * This results in two intervals: $x \geq 3$ or $x \leq -3$ . * General Rule for Quadratic Inequalities: * If $x^2 \geq a^2$ , then $x \leq -a$ or $x \geq a$ . * If $x^2 \leq a^2$ , then $-a \leq x \leq a$ . * Range Analysis: Since the square root of a real number is non-negative, the range of $f(g(x))$ is $[0, \infty)$ .
Analyzing $g(f(x))$ : * Expression: $g(\sqrt{x - 5}) = (\sqrt{x - 5})^2 - 4 = (x - 5) - 4 = x - 9$ . * Domain Analysis: Although the final expression $x - 9$ is linear and suggests a domain of all real numbers, it originated from the inner function $f(x) = \sqrt{x - 5}$ . * The restriction from the inner function ( $x - 5 \geq 0$ ) dictates the domain: $x \geq 5$ . * Range Analysis: Since the domain starts at $x = 5$ , the range starts at the corresponding $y$ value: $5 - 9 = -4$ . Thus, the range is $[-4, \infty)$ .

Inverse Functions and One-to-One Mapping

Definition of a Function (Restated): For each unique input, there is exactly one output. Mapping diagrams demonstrate this when only one line originates from each input value.
One-to-One Functions: * A function is one-to-one if no two different inputs produce the same output (no two $y$ -coordinates are identical). * Formally: $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$ . * Example: $y = x^2$ is NOT one-to-one over its entire domain because both $x = 2$ and $x = -2$ map to $y = 4$ .
Vertical vs. Horizontal Line Tests: * Vertical Line Test: Checks if a graph represents a function. If any vertical line intersects the graph more than once, it is not a function. * Horizontal Line Test: Checks if a function is one-to-one. If any horizontal line intersects the graph more than once, it is not one-to-one and therefore its inverse over that domain is not a function.
Existence of an Inverse: * An inverse function $f^{-1}$ only exists if the original function is one-to-one. * If a function fails the horizontal line test (like $y = x^2$ ), the domain must be restricted to allow for an inverse. * Restriction 1: $[0, \infty)$ resulting in the positive square root. * Restriction 2: $(-\infty, 0]$ resulting in the negative square root.
Graphical Properties of Inverses: * An inverse function is the reflection of the original function through the line $y = x$ . * When reflecting, the x and y coordinates of all points swap: $(a, b)$ on $f$ becomes $(b, a)$ on $f^{-1}$ . * Domain and Range swap between inverse functions. * Classic Example: The exponential function $y = e^x$ and the natural logarithm function $y = \ln(x)$ are inverses of each other. Both are one-to-one over their entire domains and reflect across $y = x$ .