Differential Calculus: Functions and Their Properties Lecture Notes

Definition and Notation of Functions

• Definition of a Function: A function is a specific type of relation that assigns every element from a set $A$ (referred to as the domain) to exactly one element in a set $B$ (referred to as the codomain).
• Mathematical Notation:   - The mapping is denoted as $f : A \rightarrow B$.   - The general form is $y = f(x)$.   - In this expression, $x$ is the input and $y$ is the output.
• Evaluative Example:   - Given the function: $f(x) = x^2 - 2x + 3$   - If we evaluate the function at $x = 2$:   - $f(2) = 2^2 - 2(2) + 3$   - $f(2) = 4 - 4 + 3$   - $f(2) = 3$

Independent and Dependent Variables

• Independent Variable: This represents the input of the function, typically denoted as $x$. It is the value you choose to put into the function.
• Dependent Variable: This represents the output of the function, denoted as $y$ or $f(x)$. Its value depends entirely on the choice of the independent variable.
• Function Example:   - In the function $g(t) = 5t + 2$:   - $t$ is the independent variable.   - $g(t)$ is the dependent variable.
• Real-Life Contextual Example: The temperature ($T$) of a city measured over time ($h$):   - Formula: $T = f(h)$   - In this scenario, time in hours ($h$) is the independent variable.   - The temperature ($T$) is the dependent variable because it depends on at what time of day it is measured.

Graphical Representations of Functions

• Visualization Purpose: Graphing a function allows for the understanding of its behavior, including its domain, range, intercepts, and symmetry.
• Common Function Types and Their Graphs:   - Linear Function: $f(x) = 2x + 1$. The graph is a straight line.   - Quadratic Function: $f(x) = x^2 - 4$. The graph is a parabola.   - Reciprocal Function: $f(x) = \frac{1}{x}$. The graph exhibits asymptotic behavior where it never touches the axes.

The Vertical Line Test

• Definition: The vertical line test is a visual method used to determine if a graph represents a function.
• The Rule: A graph represents a function if and only if no vertical line intersects the graph more than once.
• Failure Condition: If any vertical line passes through the graph at multiple points, it fails the test and is classified as a relation, not a function.
• Comparison Examples:   - Is a Function: $y = x^2$ is a function because any vertical line crosses the parabola at most once.   - Is Not a Function: $x^2 + y^2 = 4$ represents a circle. It is not a function because a vertical line can cross the circle at two distinct points.

Domain and Range

• Domain: This is the set of all possible input values ($x$) for which the function $f(x)$ is mathematically defined.
• Range: This is the set of all possible output values ($f(x)$) that the function can produce.
• Detailed Example Analysis: Consider the function $f(x) = \frac{1}{x - 2}$.   - Domain Computation: The function becomes undefined if the denominator is zero. Since $x - 2 = 0$ when $x = 2$, the value $2$ must be excluded.   - Domain: $(-\infty, 2) \cup (2, \infty)$.   - Range Computation: The function $f(x) = \frac{1}{x - 2}$ will never equal zero because there is no value of $x$ that can make the fraction zero (since the numerator is a constant $1$).   - Range: $(-\infty, 0) \cup (0, \infty)$.

Absolute Value and Piecewise Functions

• Absolute Value Function:   - Definition:     $f(x) = \begin{cases}x, & x \geq 0 \\ -x, & x < 0\end{cases}$   - Characteristics: The function is always non-negative ($f(x) \geq 0$). The graph forms a "V" shape centered at the origin.
• Piecewise Functions:   - Definition: A function that is defined by different expressions depending on the interval of the input value $x$.   - Example Piecewise Expression:     $f(x) = \begin{cases}x^2, & x < -2 \\ 2x + 3, & -2 \leq x < 4 \\ 5, & x \geq 4\end{cases}$   - Interpretation of the Example:     - If the input is less than $-2$, the output is calculated using the square of the input.     - if the input is between $-2$ (inclusive) and $4$ (exclusive), the output follows the linear path $2x + 3$.     - If the input is $4$ or greater, the function remains constant at an output value of $5$.

Composition of Functions

• Definition: The composition of two functions $f$ and $g$ is an operation where one function is applied to the result of another.
• Notation: $(f \circ g)(x) = f(g(x))$
• Applied Example:   - Let $f(x) = x^2 + 1$ and $g(x) = x + 3$.   - The composition $(f \circ g)(x) = f(g(x)) = (x + 3)^2 + 1$.

Problem Sets and Exercises

Exercises: Vertical Line Test and Function Identification

1. Determine whether the relation $x^2 + y^2 = 9$ is a function using the vertical line test.
2. Identify which of the following expressions define a function:    - (a) $y^2 = x$    - (b) $y = |x - 3|$    - (c) $y = \sqrt{x}$
3. Sketch the graph of the function $f(x) = x^3 - 3x$ and apply the vertical line test to confirm its status.
4. Provide an explanation for why every linear equation in the form $y = mx + b$ defines a function.

Exercises: Domain and Range Determination

1. Find the domain and range of: $f(x) = \sqrt{x - 1}$.
2. Determine the domain of: $g(x) = \frac{2x + 1}{x^2 - 4}$.
3. Find the domain and range of: $h(x) = \sqrt{\frac{1}{x - 3}}$.
4. Find the domain and range of: $f(x) = 3x^2 - 2$.
5. Find the natural domain for the following functions:    - (a) $f(x) = x^2$    - (b) $f(x) = x - 6$    - (c) $f(x) = \frac{3 - x}{(2 - x)(2x + 1)}$    - (d) $f(x) = \frac{2}{x(x + 5)(x - 1)}$    - (e) $f(x) = \sqrt{x}$    - (f) $f(x) = 3 + \sqrt{x}$    - (g) $f(x) = \sqrt{x + 1}$    - (h) $f(x) = 1 - \sqrt{x - 1}$    - (i) $f(x) = \sqrt{x^2 - x - 2}$    - (j) $f(x) = \sqrt{x^2 - 9x + 18}$    - (k) $f(x) = \sqrt{x^2 + 2x - 15}$    - (l) $f(x) = \sqrt{(x^2 + 2x)(x - 2)}$    - (m) $f(x) = \frac{x + 4}{3^2 - 16}$    - (n) $f(x) = \frac{x - 1}{x^2 + 2x - 3}$

Exercises: Absolute Value and Piecewise Functions

1. Solve for $x$ given the absolute value equation: $f(x) = |2x - 5| = 3$.
2. Identify the vertex and sketch the graph for: $g(x) = |x - 4| - 2$.
3. Use the piecewise function below to answer the following:    $f(x) = \begin{cases}2x + 1, & x < 3 \\ x^2 - 4, & x \geq 3\end{cases}$    - Calculate $f(1)$, $f(4)$, $f(-3)$, and $f(0)$.    - Solve for $x$ such that $f(x) = 5$.

Exercises: Composition of Functions

1. Given $f(x) = x^2 - 3x + 2$ and $g(x) = x - 1$, find $(f \circ g)(x)$.
2. Given $p(x) = 2x + 5$ and $q(x) = \sqrt{x + 4}$, find $(p \circ q)(x)$.
3. Given $f(x) = \frac{1}{x}$ and $g(x) = x + 3$, find the expression and the domain for $(g \circ f)(x)$.
4. Given $f(x) = x^2 - 4$ and $g(x) = 2x + 1$:    - Compute $(f \circ g)(2)$.    - Compute $(g \circ f)(2)$.
5. Let $f(x) = x - 3$ and $g(x) = x + 1$. Find $(f \circ g)(x)$.
6. Let $f(x) = \sqrt{x - 2}$ and $g(x) = x^2 + 4$. Find $(g \circ f)(x)$.
7. Let $f(x) = 2x^3 - 6$ and $g(x) = \sqrt[3]{x + 3}$. Find $(f \circ g)(x)$.

Exercises: Graphing and Classification

Sketch the graphs of the following and determine if they represent a function or a relation:

• (a) $y = |x - 1|$
• (b) $y = -|x|$
• (c) $3x + y = 0$
• (d) $x - 5y = 10$
• (e) $3x = 6$
• (f) $y = -4$
• (g) $y = \sqrt{x - 1}$
• (h) $y = -\sqrt{x + 1}$
• (i) $y = 1 + \sqrt{x}$
• (j) $x + y^2 = 3$
• (k) $y^2 = x - 2$
• (l) $(y - 2)^2 = x$
• (m) $(y + 1)^2 = x + 1$
• (n) $(x - 1)^2 = 1 - y$

1. The relation $x^2 + y^2 = 9$ is not a function using the vertical line test because a vertical line can intersect the circle at two points.

2. The expressions that define a function are:

   • (b) $y = |x - 3|$ (this gives one output for each input)
   • (c) $y = \frac{1}{ ext{input}}$ (this gives one output for each input)

   (a) $y^2 = x$ does not define a function because it can give two outputs for a single input (both positive and negative values of $y$).

3. The graph of the function $f(x) = x^3 - 3x$