Functions are the building blocks of calculus. They describe relationships between variables, allowing us to model real-world phenomena. Understanding function notation, types, and evaluation is crucial for solving complex problems in calculus.

Domain and range define where functions operate. Graphing functions reveals their behavior, including key features like intercepts and asymptotes. Identifying zeros helps solve equations and understand function behavior, setting the stage for more advanced calculus concepts.

Function Fundamentals

Function notation evaluation

• Function notation $f(x)$ represents the output value when the input is $x$
  • Evaluate $f(a)$ by substituting $a$ for $x$ in the function's equation and simplify
• Function types and their notation
  • Linear functions $f(x) = mx + b$ with slope $m$ and y-intercept $b$
  • Quadratic functions $f(x) = ax^2 + bx + c$ with constants $a$, $b$, and $c$
  • Polynomial functions $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ with constants $a_i$
  • Rational functions $f(x) = \frac{P(x)}{Q(x)}$ with polynomial functions $P(x)$ and $Q(x)$
  • Exponential functions $f(x) = a^x$ with positive constant $a$
  • Logarithmic functions $f(x) = \log_a(x)$ with positive constant $a \neq 1$
  • Trigonometric functions $\sin(x)$, $\cos(x)$, $\tan(x)$, etc.
  • Piecewise functions defined by different equations over different intervals of the domain

Domain and range identification

• Domain: Set of all possible input values (usually $x$) for a function
  • Most functions have a domain of all real numbers, unless restricted
    • Rational functions exclude $x$ values that make the denominator zero
    • Square root functions require a non-negative argument under the square root
    • Logarithmic functions require a positive argument
• Range: Set of all possible output values (usually $y$) for a function
  • Determined by the function's equation and domain restrictions
    • Linear functions have a range of all real numbers
    • Quadratic functions have a range depending on the parabola direction (up or down)
    • Exponential functions always have a positive range
    • Logarithmic functions have a range of all real numbers
    • Trigonometric functions have a limited range (e.g., $-1 \leq \sin(x) \leq 1$)

Function graphing and features

• Key features of function graphs
  • x-intercepts (zeros) where the graph crosses the x-axis ($y = 0$)
  • y-intercept where the graph crosses the y-axis ($x = 0$)
  • Symmetry across the y-axis (even functions), origin (odd functions), or other lines
  • Asymptotes that the graph approaches as $x$ or $y$ approaches infinity or a specific value
    • Vertical asymptotes occur at $x$ values where the function is undefined (e.g., denominators equal to zero)
    • Horizontal asymptotes occur when the function approaches a constant value as $x$ approaches positive or negative infinity
  • Intervals of increase/decrease where the function is increasing or decreasing
  • Local maxima/minima where the function reaches a highest or lowest value within a specific interval
  • Concavity indicating the direction of the curve (upward or downward)
  • Inflection points where the concavity changes

Zeros of functions

• Zeros (roots) of a function are the $x$ values where $f(x) = 0$
  • Algebraic methods
    1. Factoring: Factor the function and set each factor equal to zero
    2. Quadratic formula: For quadratic functions, use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
    3. Rational root theorem: For polynomial functions, list potential rational zeros and test using substitution
  • Graphical methods
    • Identify the x-intercepts of the function's graph

Function Representations and Operations

Function representations

• Tables list input ($x$) and output ($y$ or $f(x)$) values in two columns
  • Identify patterns in the table to determine the function type
• Graphs plot points ($x$, $y$) on a coordinate plane and connect them to create a curve or line
  • Analyze the graph's key features to identify the function type
• Equations express the relationship between the input and output using mathematical symbols
  • Identify the function type based on the equation's form (linear, quadratic, exponential)

Function operations and composition

• Arithmetic operations on functions
  • Addition $(f + g)(x) = f(x) + g(x)$
  • Subtraction $(f - g)(x) = f(x) - g(x)$
  • Multiplication $(f \cdot g)(x) = f(x) \cdot g(x)$
  • Division $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ where $g(x) \neq 0$
• Composition of functions combines two or more functions by using the output of one as the input of another
  • $(f \circ g)(x) = f(g(x))$: First, evaluate $g(x)$, then use the result as the input for $f$
  • Domain of the composite function is the domain of $g$ restricted to values that produce outputs within the domain of $f$

Symmetry in functions

• Even functions are symmetric about the y-axis
  • Definition: $f(-x) = f(x)$ for all $x$ in the domain
  • Examples: $f(x) = x^2$, $f(x) = \cos(x)$
• Odd functions are symmetric about the origin
  • Definition: $f(-x) = -f(x)$ for all $x$ in the domain
  • Examples: $f(x) = x^3$, $f(x) = \sin(x)$
• Functions that are neither even nor odd have no symmetry or are symmetric about other lines (e.g., $y = x$)
  • Example: $f(x) = x^2 + x$
• Implications on graphs
  • Even functions: If $(a, b)$ is on the graph, then $(-a, b)$ is also on the graph
  • Odd functions: If $(a, b)$ is on the graph, then $(-a, -b)$ is also on the graph

Function Transformations and Continuity

• Function transformations alter the graph of a function
  • Vertical shifts: $f(x) + k$ moves the graph up $k$ units
  • Horizontal shifts: $f(x - h)$ moves the graph right $h$ units
  • Vertical stretches/compressions: $af(x)$ stretches ($|a| > 1$) or compresses ($0 < |a| < 1$) vertically
  • Horizontal stretches/compressions: $f(bx)$ stretches ($0 < |b| < 1$) or compresses ($|b| > 1$) horizontally
  • Reflections: $-f(x)$ reflects over the x-axis, $f(-x)$ reflects over the y-axis
• Continuous functions have no breaks, holes, or jumps in their graphs
  • A function $f(x)$ is continuous at a point $a$ if:
    1. $f(a)$ is defined
    2. $\lim_{x \to a} f(x)$ exists
    3. $\lim_{x \to a} f(x) = f(a)$
  • A function is continuous on an interval if it is continuous at every point in that interval
• Discontinuous functions have at least one point where continuity conditions are not met
  • Types of discontinuities:
    • Removable (point) discontinuity: A hole in the graph
    • Jump discontinuity: The function "jumps" from one value to another
    • Infinite discontinuity: The function approaches infinity as x approaches a certain value
• Inverse functions "undo" the original function, swapping input and output
  • For a function $f$, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
  • The graph of an inverse function is the reflection of the original function over the line $y = x$