Functions and Graphs Overview

Functions

• Definition: A function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output.
• Notation: Typically expressed as f(x), where f is the function name and x is the input.
• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Types of Functions:
  • Linear: f(x) = mx + b (m = slope, b = y-intercept)
  • Quadratic: f(x) = ax² + bx + c (parabolic shape)
  • Polynomial: f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0
  • Exponential: f(x) = a * b^x (where b > 0, b ≠ 1)
  • Logarithmic: f(x) = log_b(x) (inverse of exponential)
  • Trigonometric: Includes functions like sine, cosine, and tangent.

Graphs

• Coordinate System: Functions are typically graphed on a Cartesian plane (x-y plane).
• Axes:
  • X-axis (horizontal)
  • Y-axis (vertical)
• Plotting Points: Each point is represented as (x, f(x)).
• Intercepts:
  • X-intercept: The point(s) where the graph crosses the x-axis (f(x) = 0).
  • Y-intercept: The point where the graph crosses the y-axis (x = 0).
• Slope:
  • Represents the rate of change.
  • For linear functions, slope (m) indicates steepness: positive (upward), negative (downward).
• Asymptotes: Lines that the graph approaches but never touches.
  • Vertical Asymptotes: x = a (undefined f(x)).
  • Horizontal Asymptotes: y = b (value f(x) approaches as x → ±∞).

Transformations of Functions

• Vertical Shifts: f(x) + k (up if k > 0, down if k < 0).
• Horizontal Shifts: f(x - h) (right if h > 0, left if h < 0).
• Reflections:
  • Across x-axis: -f(x)
  • Across y-axis: f(-x)
• Stretching and Compressing:
  • Vertical Stretch/Compression: af(x) (a > 1 = stretch, 0 < a < 1 = compression).
  • Horizontal Stretch/Compression: f(bx) (0 < b < 1 = stretch, b > 1 = compression).

Compositions and Inverses

• Composition of Functions: (f ∘ g)(x) = f(g(x)), combining two functions.
• Inverse Functions:
  • Denoted as f⁻¹(x), swaps input and output.
  • Finding Inverses: Solve for x in terms of y and switch x and y.

Key Concepts

• Continuity: A function is continuous if there are no breaks in the graph.
• Increasing/Decreasing Functions:
  • Increasing: f(x₁) < f(x₂) for x₁ < x₂.
  • Decreasing: f(x₁) > f(x₂) for x₁ < x₂.
• Maximum/Minimum Values: Highest or lowest points on the graph, important for optimization problems.
• End Behavior: Behavior of the graph as x approaches ±∞, important for polynomial functions.

Functions

• A function defines a consistent relationship between inputs and outputs, ensuring each input correlates to one unique output.
• Commonly notated as f(x), with f representing the function and x indicating the input value.
• The domain encompasses all possible input values, while the range includes all possible output values.

Types of Functions

• Linear Function: Expressed as f(x) = mx + b, with m as the slope and b as the y-intercept.
• Quadratic Function: Defined by f(x) = ax² + bx + c, forming a parabolic curve.
• Polynomial Function: General form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer.
• Exponential Function: Given by f(x) = a * b^x, for b > 0 and b ≠ 1.
• Logarithmic Function: Written as f(x) = log_b(x), acting as the inverse of the exponential function.
• Trigonometric Functions: Include sine, cosine, and tangent, essential in circular motion and oscillatory phenomena.

Graphs

• Functions are typically displayed on a Cartesian plane, with an x-axis (horizontal) and y-axis (vertical).
• Each point on the graph corresponds to (x, f(x)), representing the input-output pairing.
• Intercepts:
  • X-Intercept: Occurs where the graph intersects the x-axis (f(x) = 0).
  • Y-Intercept: Occurs where the graph intersects the y-axis (x = 0).
• Slope: Indicates the function's rate of change; for linear functions, a positive slope signifies an upward trend, while a negative slope indicates a downward trend.
• Asymptotes: Lines that the graph approaches but never crosses; vertical asymptotes exist where f(x) becomes undefined, while horizontal asymptotes are the value f(x) approaches as x tends to infinity.

Transformations of Functions

• Vertical Shifts: Represented as f(x) + k, moving the graph upward (k > 0) or downward (k < 0).
• Horizontal Shifts: Denoted by f(x - h), shifting right (h > 0) or left (h < 0).
• Reflections:
  • Across the x-axis: Implemented by -f(x).
  • Across the y-axis: Implemented by f(-x).
• Stretching and Compressing:
  • Vertical alterations: af(x) facilitates stretching (a > 1) or compressing (0 < a < 1).
  • Horizontal adjustments: f(bx) where a value between 0 and 1 indicates stretching, and a value greater than 1 indicates compressing.

Compositions and Inverses

• Composition of Functions: Expressed as (f ∘ g)(x) = f(g(x)), combining functions to create a new output.
• Inverse Functions: Noted as f⁻¹(x), reversing the roles of input and output.
• Finding Inverses: Involves solving for x in terms of y, followed by switching x and y.

Key Concepts

• Continuity: A function is continuous when its graph has no breaks or gaps.
• Increasing/Decreasing Functions:
  • Increasing: If f(x₁) < f(x₂) for x₁ < x₂.
  • Decreasing: If f(x₁) > f(x₂) for x₁ < x₂.
• Maximum/Minimum Values: Critical points that represent the highest or lowest output values, important in optimization.
• End Behavior: Describes how the graph behaves as x approaches positive or negative infinity, particularly relevant for polynomials.

Definition of a Function

• A function consists of ordered pairs where each x-value corresponds to exactly one y-value.
• Vertical line test verifies if a relation is a function; if a vertical line intersects the graph more than once, it is not a function.

Examples of Functions

• First example: {(2,3), (3,5), (6,5), (6,3), (8,10) } - Not a function since x=6 has two different y-values (5 and 3).
• Second example: {(-1,7), (-6,8), (1,4), (4,11), (5,11)} - This is a function; each x-value appears only once.

Function Notation

• Function notation replaces traditional "y=" with "f(x)", pronounced as "f of x".
• Represents the value of the function at the given x.
• Variables other than f can be used, such as g or h.

Evaluating Functions

• To find the value of f(x) at a specific x-value, substitute x in the function's equation.
• Example:
  • For f(x) = 3x - 15, evaluating at x=3 gives f(3) = 3(3) - 15 = -6.
  • Find f(7) when f(x) = (1/7)x² + 5 requires substituting 7 into the formula.

Solving for x in Function Models

• For g(x) = -8x - 2 and g(x) = -26, solve for x by equating and isolating x.
• For h(x) = 3(2x - 4) + 2 and h(x) = 20, use similar methods to find x.

Function Graph Interpretation

• Graphs can represent functions, allowing for visual assessment of values at specific x-values.
• Determine function outputs for specific inputs, e.g., f(-8), f(6), and identify when f(x) equals a certain number.

Coordinate Points from Function Statements

• Translate function statements into coordinate notation:
  • 𝑓(5) = −1 translates to the point (5, -1).
  • 𝑔(−2) = 0 translates to (-2, 0).
  • ℎ(6) = 6 translates to (6, 6).

Definition of a Function

• A function consists of ordered pairs where each x-value corresponds to exactly one y-value.
• Vertical line test verifies if a relation is a function; if a vertical line intersects the graph more than once, it is not a function.

Examples of Functions

• First example: {(2,3), (3,5), (6,5), (6,3), (8,10) } - Not a function since x=6 has two different y-values (5 and 3).
• Second example: {(-1,7), (-6,8), (1,4), (4,11), (5,11)} - This is a function; each x-value appears only once.

Function Notation

• Function notation replaces traditional "y=" with "f(x)", pronounced as "f of x".
• Represents the value of the function at the given x.
• Variables other than f can be used, such as g or h.

Evaluating Functions

• To find the value of f(x) at a specific x-value, substitute x in the function's equation.
• Example:
  • For f(x) = 3x - 15, evaluating at x=3 gives f(3) = 3(3) - 15 = -6.
  • Find f(7) when f(x) = (1/7)x² + 5 requires substituting 7 into the formula.

Solving for x in Function Models

• For g(x) = -8x - 2 and g(x) = -26, solve for x by equating and isolating x.
• For h(x) = 3(2x - 4) + 2 and h(x) = 20, use similar methods to find x.

Function Graph Interpretation

• Graphs can represent functions, allowing for visual assessment of values at specific x-values.
• Determine function outputs for specific inputs, e.g., f(-8), f(6), and identify when f(x) equals a certain number.

Coordinate Points from Function Statements

• Translate function statements into coordinate notation:
  • 𝑓(5) = −1 translates to the point (5, -1).
  • 𝑔(−2) = 0 translates to (-2, 0).
  • ℎ(6) = 6 translates to (6, 6).