Thomas’ Calculus Fifteenth Edition Chapter 1 - Functions

Functions; Domain and Range

• Definition of a Function: A function $f$ from a set $D$ to a set $Y$ is a rule that assigns a single value in $Y$ to each $x$ in $D$.
• Notation: The relationship is expressed as $y = f(x)$, read as "$y$ equals $f$ of $x$."
• Variables:
  • The symbol $f$ represents the function itself.
  • The letter $x$ is the independent variable, representing the input value of $f$.
  • The letter $y$ is the dependent variable or output value of $f$ at $x$.
• Machine Metaphor: A function can be visualized as a machine where an input $x$ enters, and a unique output $f(x)$ is produced.
• Mapping: A function assigns a unique element of $Y$ to each element in $D$. It is possible for different inputs to map to the same output, but one input cannot map to multiple different outputs.

Standard Functions: Domain and Range Examples

• Square Function: $y = x^2$
  • Domain ($x$): $(-\infty, \infty)$
  • Range ($y$): $[0, \infty)$
• Reciprocal Function: $y = \frac{1}{x}$
  • Domain ($x$): $(-\infty, 0) \cup (0, \infty)$
  • Range ($y$): $(-\infty, 0) \cup (0, \infty)$
• Square Root Function: $y = \sqrt{x}$
  • Domain ($x$): $[0, \infty)$
  • Range ($y$): $[0, \infty)$
• Shifted Square Root: $y = \sqrt{4 - x}$
  • Domain ($x$): $(-\infty, 4]$
  • Range ($y$): $[0, \infty)$
• Semicircle Function: $y = \sqrt{1 - x^2}$
  • Domain ($x$): $[-1, 1]$
  • Range ($y$): $[0, 1]$

Graphs of Functions and the Vertical Line Test

• Definition of a Graph: The graph of a function $f$ is the set of points $(x, y)$ in the coordinate plane for which $y = f(x)$.
• Geometric Interpretation: If a point $(x, y)$ lies on the graph, the value $f(x)$ represents the height of the graph above or below the point $x$ on the horizontal axis. A negative value indicates the graph is below the axis.
• The Vertical Line Test:
  • A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once.
  • This confirms that each $x$ in the domain corresponds to exactly one value $f(x)$.
  • If $a$ is in the domain, the vertical line $x = a$ intersects the graph at exactly one point $(a, f(a))$.
• Examples of the Test:
  • A circle is not the graph of a function because it fails the vertical line test.
  • The upper semicircle, defined by $f(x) = \sqrt{1 - x^2}$, is a function.
  • The lower semicircle, defined by $g(x) = -\sqrt{1 - x^2}$, is also a function.

Piecewise-Defined Functions

• Definition: Functions that apply different formulas to different parts of their domain.
• Absolute Value Function:
  • Defined as follows:
  • $|x| = x$ if $x \ge 0$
  • $|x| = -x$ if $x < 0$
  • Domain: $(-\infty, \infty)$
  • Range: $[0, \infty)$
• Greatest Integer Function (Floor Function):
  • Symbolized as $y = \lfloor x \rfloor$.
  • This function outputs the greatest integer less than or equal to $x$.
  • Its graph lies on or below the line $y = x$, serving as an "integer floor."
• Least Integer Function (Ceiling Function):
  • Symbolized as $y = \lceil x \rceil$.
  • This function outputs the smallest integer greater than or equal to $x$.
  • Its graph lies on or above the line $y = x$, serving as an "integer ceiling."

Function Behavior: Increasing, Decreasing, and Symmetry

• Increasing and Decreasing Functions:
  • Let $f$ be a function on an interval $I$ and $x_1, x_2$ be points in $I$.
  • Increasing: $f$ is increasing on $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.
  • Decreasing: $f$ is decreasing on $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.
• Even and Odd Functions:
  • Even Function: A function is even if $f(-x) = f(x)$ for every $x$ in the domain. The graph is symmetric about the y-axis (e.g., $y = x^2$).
  • Odd Function: A function is odd if $f(-x) = -f(x)$ for every $x$ in the domain. The graph is symmetric about the origin (e.g., $y = x^3$).
• Effect of Constant Terms:
  • Adding a constant to an even function (e.g., $y = x^2 + 1$) preserves its even property and y-axis symmetry.
  • Adding a constant to an odd function (e.g., $y = x + 1$) typically results in a function that is neither even nor odd, as origin symmetry is lost.

Catalog of Common Functions

• Linear Functions: Functions of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Proportionality: Two variables are proportional if $y = kx$ for some nonzero constant $k$.
• Power Functions: Functions of the form $f(x) = x^a$.
  • Examples include integer powers ($x^n$), reciprocal powers ($x^{-1}, x^{-2}$), and root functions ($x^{1/2}, x^{2/3}$).
• Polynomial Functions: A function $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer and coefficients are real constants.
• Rational Functions: A quotient of two polynomials, $f(x) = \frac{p(x)}{q(x)}$. The domain is all real numbers where the denominator $q(x) \ne 0$. Graphs often approach lines called asymptotes.
• Algebraic Functions: Functions constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and roots).
• Transcendental Functions: Functions that are not algebraic, including:
  • Trigonometric and inverse trigonometric functions.
  • Exponential functions.
  • Logarithmic functions.

Combining Functions

• Arithmetic Operations:
  • Sum: $(f + g)(x) = f(x) + g(x)$
  • Difference: $(f - g)(x) = f(x) - g(x)$
  • Product: $(f \cdot g)(x) = f(x) \cdot g(x)$
  • Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$
• Domain of Combined Functions: For sums, differences, and products, the domain is the intersection of the domains of $f$ and $g$ ($D_f \cap D_g$). For quotients, the domain is the intersection EXCEPT where the denominator $g(x) = 0$.
• Composite Functions:
  • Formula: $(f \circ g)(x) = f(g(x))$.
  • Process: The output of the first function $g$ is used as the input for the second function $f$.
  • Domain: Consists of values $x$ in the domain of $g$ such that the output $g(x)$ lies within the domain of $f$.

Shifting, Scaling, and Reflecting Graphs

• Vertical Shifts:
  • $y = f(x) + k$ shifts the graph up if $k > 0$ and down if $k < 0$.
• Horizontal Shifts:
  • $y = f(x + h)$ shifts the graph left if $h > 0$ and right if $h < 0$.
• Vertical Scaling (for $c > 1$):
  • Stretch: $y = cf(x)$.
  • Compression: $y = \frac{1}{c}f(x)$.
• Horizontal Scaling (for $c > 1$):
  • Compression: $y = f(cx)$.
  • Stretch: $y = f\left(\frac{x}{c}\right)$.
• Reflections:
  • Across x-axis: $y = -f(x)$.
  • Across y-axis: $y = f(-x)$.

Trigonometric Functions: Fundamentals

• Radian Measure: The radian measure of a central angle is $\theta = \frac{s}{r}$. For a unit circle ($r = 1$), the angle equals the arc length ($\theta = s$).
• Standard Position: Angles in the xy-plane are measured from the positive x-axis. Positive measures are counter-clockwise; negative measures are clockwise.
• The Six Basic Functions:
  • $\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$
  • $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$
  • $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$
  • $\csc(\theta) = \frac{\text{hyp}}{\text{opp}}$
  • $\sec(\theta) = \frac{\text{hyp}}{\text{adj}}$
  • $\cot(\theta) = \frac{\text{adj}}{\text{opp}}$
• The ASTC Rule: "All Students Take Calculus" identifies quadrants where functions are positive:
  • Quadrant I: All functions positive.
  • Quadrant II: Sine (and csc) positive.
  • Quadrant III: Tangent (and cot) positive.
  • Quadrant IV: Cosine (and sec) positive.

Table of Key Trigonometric Values

Trigonometric Periodicity and Identities

• Periodicity: A function is periodic if $f(x + p) = f(x)$ for a positive constant $p$.
  • Period $\pi$: $\tan(x), \cot(x)$.
  • Period $2\pi$: $\sin(x), \cos(x), \sec(x), \csc(x)$.
• Odd/Even Identities:
  • Even: $\cos(-x) = \cos(x)$; $\sec(-x) = \sec(x)$.
  • Odd: $\sin(-x) = -\sin(x)$; $\tan(-x) = -\tan(x)$; $\csc(-x) = -\csc(x)$; $\cot(-x) = -\cot(x)$.
• Fundamental Identities:
  • $\cos^2(\theta) + \sin^2(\theta) = 1$
  • $1 + \tan^2(\theta) = \sec^2(\theta)$
  • $1 + \cot^2(\theta) = \csc^2(\theta)$
• Addition Formulas:
  • $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$
  • $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$
• Double-Angle and Half-Angle Formulas:
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$
  • $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$

Advanced Geometric and Trigonometric Principles

• Law of Cosines: For a triangle with sides $a, b, c$ and angle $\theta$ opposite side $c$:
  • $c^2 = a^2 + b^2 - 2ab\cos(\theta)$
• Special Trig Inequalities:
  • $-|\theta| \le \sin(\theta) \le |\theta|$
  • $-|\theta| \le 1 - \cos(\theta) \le |\theta|$
• General Sinusoid Formula:
  • $y = a f(b(x + c)) + d$
  • $a$: Vertical stretch/compression (amplitude if sine/cosine).
  • $b$: Horizontal stretch/compression (affects period).
  • $c$: Horizontal shift.
  • $d$: Vertical shift.