University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2023
Views:

139

Pages:

4
Author:

Niklas

III. Relations and Functions (Ch. 1-481-5) I. DOMAIN & RANGE: Domain: ask yourself is there any value of x that I could not use? y = 3x + 7 2. y = 5/x domain: (-infinity, infinity) domain: (-infinity, 0) U (0, infinity) range: (-infinity, infinity) range: (-infinity, 0) U (0, infinity) Range: ask yourself is there any value of y that I would not get using the domain? y = x^2 4. y = sqrt(x) domain: (-infinity, infinity) domain: [0, infinity) range: [0, infinity) range: [0, infinity) II. EVEN & ODD FUNCTIONS: even functions have line symmetry. odd functions have point symmetry. Determine whether each function is odd or even by the equation y = x^3 + 3x 6. y = x^4 - 2x^2 7. y = x^3 + 4x^2 + x odd even neither Now determine by looking at the graph of the function: III. PARENT FUNCTIONS linear function DOMAIN: (-∞, ∞) y = x RANGE: (-∞, ∞) Y-INT: (0) X-INT: (0) INC: (-∞, ∞) DEC: none CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → -∞, y → -∞ absolute value function y = |x| DOMAIN: (-∞, ∞) RANGE: [0, ∞) Y-INT: [0] X-INT: (0) INC: (0, ∞) DEC: (-∞, 0) CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → -∞, y → ∞ quadratic function DOMAIN: (-∞, ∞) RANGE: [0, ∞) Y-INT: [0] X-INT: (0, 0) INC: (0, ∞) DEC: (-∞, 0) CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → -∞, y → ∞ y = x² Square root function y = sqrt(x) DOMAIN: [0, ∞) RANGE: [0, ∞) Y-INT: [0] X-INT: (0) INC: (0, ∞) DEC: none CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → 0, y → 0 Cubic function y = x^3 DOMAIN: (-∞, ∞) RANGE: (-∞, ∞) Y-INT: [0] X-INT: (0) INC: (-∞, ∞) DEC: none CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → -∞, y → -∞ Cube root function y = x^(1/3) DOMAIN: (-∞, ∞) RANGE: (-∞, ∞) Y-INT: [0] X-INT: (0) INC: (-∞, ∞) DEC: none CON: none END BEHAVIOR: as x → ∞, y → ∞ as x → -∞, y → -∞ NOTE: almost all of the important functions are continuous all of the important functions have two points in common: (0, 0) and (1, 1) IV. TRANSFORMATIONS OF PARENT FUNCTIONS vertical shift f(x) = a(x-h) + k if a<0, reflect over x-axis if |a|<1, shrink the graph if |a|>1, stretch the graph graph the parent function in one color, and the transformation in another: parent function: quadratic y = (x-1)^2 + 2 domain: (-∞, ∞) range: [2, ∞) parent function: absolute value y = -|x| + 1 domain: (-∞, ∞) range: (-∞, 1] write the equation of the transformed function: parent function: absolute value equation: y = |x| - 4 domain: (-∞, ∞) range: [-4, ∞) parent function: quadratic equation: y = (x-1)^2 + 3 domain: (-∞, ∞) range: [3, ∞)