Understanding Function Properties

• A function is a relation where each input has exactly one output.

Function Properties

• Domain refers to all possible input values (x-values) for which the function is defined.
• Range refers to all possible output values (y-values) that the function can produce.
• Intercepts are points where the function's graph intersects the x-axis (x-intercept) or the y-axis (y-intercept).
• Symmetry can be even (symmetric about the y-axis, f(-x) = f(x)), odd (symmetric about the origin, f(-x) = -f(x)), or neither.
• Intervals of increase and decrease: where the function's values are going up or down as x increases.
• Maximum and minimum values: the highest and lowest points of the function, either locally or globally.
• End behavior: describes what happens to the function’s values as x approaches positive or negative infinity.
• Continuity: A function is continuous if its graph can be drawn without lifting the pen, meaning there are no breaks, holes, or asymptotes.
• Discontinuity: Points where a function is not continuous. Can be removable (hole), jump, or infinite (asymptote).

Function Operations

• Addition: (f + g)(x) = f(x) + g(x)
• Subtraction: (f - g)(x) = f(x) - g(x)
• Multiplication: (f * g)(x) = f(x) * g(x)
• Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
• Scalar Multiplication: (c * f)(x) = c * f(x), where c is a constant.
• The domain of the resulting function consists of x-values that are in the domains of f and g.
• For division, we also exclude any x-values where g(x) = 0.

Function Composition

• Composition: (f ∘ g)(x) = f(g(x)), meaning apply g first, then apply f to the result.
• To evaluate, first find g(x) and then substitute that expression into f(x).
• The domain of (f ∘ g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f.
• Composition is not commutative: f(g(x)) is generally not equal to g(f(x)).

Graphing Functions

• Plotting points: calculate several points (x, f(x)) and plot them on a coordinate plane.
• Transformations: altering the graph of a function by shifting, stretching, compressing, or reflecting.
  • Vertical shift: adding a constant to the function, f(x) + c (up if c > 0, down if c < 0).
  • Horizontal shift: replacing x with (x - c) in the function, f(x - c) (right if c > 0, left if c < 0).
  • Vertical stretch/compression: multiplying the function by a constant, c * f(x) (stretch if |c| > 1, compress if 0 < |c| < 1).
  • Horizontal stretch/compression: replacing x with (c * x) in the function, f(cx) (compress if |c| > 1, stretch if 0 < |c| < 1).
  • Reflection over the x-axis: multiplying the function by -1, -f(x).
  • Reflection over the y-axis: replacing x with -x, f(-x).
• Symmetry:
  • Even functions are symmetric with respect to the y-axis.
  • Odd functions are symmetric with respect to the origin.
• Asymptotes:
  • Vertical asymptotes occur where the function approaches infinity (often where the denominator of a rational function is zero).
  • Horizontal asymptotes describe the function's behavior as x approaches infinity or negative infinity. Found by examining the limits as x goes to ±∞.
• Key functions to know:
  • Linear functions: f(x) = mx + b
  • Quadratic functions: f(x) = ax^2 + bx + c
  • Polynomial functions
  • Rational functions: p(x)/q(x) where p(x) and q(x) are polynomials.
  • Exponential functions: f(x) = a^x
  • Logarithmic functions: f(x) = log_a(x)
  • Trigonometric functions: sin(x), cos(x), tan(x)
  • Absolute value function: f(x) = |x|
  • Square root function: f(x) = √x

Explanation of function properties including domain, range, intercepts. Discussion of symmetry, intervals of increase/decrease, and maximum/minimum values. Covers end behavior, continuity and discontinuity.