Functions and Graphs Photo Guide to Key Concepts
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Summary of aspects of functions: domain, range, domain restrictions. Worked out for all elementary mathematical functions: linear, quadratic, etc.
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Functions and Graphs Photo Guide to Key Concepts
- 1. Functions & Graphs Photo: www.flickr.com
- 2. Relations Functions Functions & Graphs 2
- 3. Domain & Range 풟 = −3, −2,0,1 Domain 풟 = Set of independent variables Range ℛ = Set of dependent variables Functions & Graphs 3 ℛ = −1,1,2 풟 = ℝ ℛ = −∞, 41 2
- 4. Notation Parameters Functions & Graphs 4 푓: 푥 ⟼ 2푥2 + 5 푦 = 2푥2 + 5 푦 = 푓 푥 Argument 푓: 푥 ⟼ 푎푥2 + 푏
- 5. Domain restriction 푓 is NOT a function on Functions & Graphs 5 푓: 푥 ⟼ 1 푥 − 2 풟 = ℝ 풟 = 푥 푥 ∈ ℝ ∧ 푥 ≠ 2 푓 IS a function on
- 6. Domain by context Functions & Graphs 6 푣 푡 = 100 푡 Average velocity 푣 푡 is mathematically a function on 풟 = 푡 푡 ∈ ℝ ∧ 푡 ≠ 0 푣 푡 is by context a function on 풟 = 푡 푡 ∈ ℝ ∧ 푡 > 0 Photo: Wikipedia.org
- 7. Linear functions 푓: 푥 ⟼ 푎푥 + 푏 풟 = ℝ ℛ = ℝ Functions & Graphs 7 푏 Δ푥 Δ푦 푎 = Δ푦 Δ푥 Straight Line
- 8. Quadratic functions - I 푓: 푥 ⟼ 푎푥2 + 푏푥 + 푐 풟 = ℝ ℛ = 푦푣,∞ 푥푣, 푦푣 = − 푏 2푎 , 푓 − 푏 2푎 Functions & Graphs 8 Axis of symmetry Vertex 푎 > 0 Vertex = Parabola
- 9. Quadratic functions - II 푓: 푥 ⟼ 푎푥2 + 푏푥 + 푐 풟 = ℝ ℛ = −∞, 푦푣 푥푣, 푦푣 = − 푏 2푎 , 푓 − 푏 2푎 Functions & Graphs 9 Vertex Axis of symmetry 푎 < 0 Vertex = Parabola
- 10. Quadratic functions – Determine vertex Functions & Graphs 10 푦 = −1 2푥2 + 4푥 + 6 푥푣, 푦푣 = 4,14 ⇒ ℛ = −∞, 14 Vertex Axis of symmetry Vertex = Parabola “Complete squares” 푦 = −1 2 푥2 − 8푥 − 12 푦 = −1 2 푥 − 4 2 − 16 − 12 푦 = −1 2 푥 − 4 2 − 28 divide by 1st parameter Include half of 2nd parameter in the square Compensate for the extra constant Square term smallest (0), if 푥 = 푥푣 = 4 푦푣 = −1 2 푥푣 − 4 2 − 28 = 14
- 11. Radical & Absolute value functions Functions & Graphs 11 푓: 푥 ⟼ 4푥 − 3 풟 = 3 4 ,∞ ℛ = 0,∞ 푓: 푥 ⟼ 푥 풟 = ℝ ℛ = 0,∞
- 12. Reciprocal & Rational functions Functions & Graphs 12 푓: 푥 ⟼ 1 푥 풟 = 푥 푥 ∈ ℝ ∧ 푥 ≠ 0 ℛ = 푦 푦 ∈ ℝ ∧ 푦 ≠ 0 푓: 푥 ⟼ 3푥 − 4 2푥 + 5 풟 = 푥 푥 ∈ ℝ ∧ 푥 ≠ −21 2 ℛ = 푦 푦 ∈ ℝ ∧ 푦 ≠ 11 2 Hyperbola Asymptotes
- 13. Rational functions Asymptotes 푓: 푥 ⟼ 3푥 − 4 2푥 + 5 풟 = 푥 푥 ∈ ℝ ∧ 푥 ≠ −21 2 ℛ = 푦 푦 ∈ ℝ ∧ 푦 ≠ 11 2 Functions & Graphs 13 Finding the vertical asymptote Fraction is undefined, if denominator = 0 2푥 + 5 = 0 ⇒ 푥 = −21 2 Finding the horizontal asymptote Take a ‘huge’ number for 푥 푦 = 3 ∙ 10100 − 4 2 ∙ 10100 + 5 ≈ 3 ∙ 10100 2 ∙ 10100 = 3 2 = 1 1 2
- 14. Many-to-one vs. One-to-one Functions & Graphs 14
- 15. Even vs. Odd 푓 −푥 = 푓 푥 푓 −푥 = −푓 푥 For all 푥 ∈ 풟 For all 푥 ∈ 풟 Functions & Graphs 15
- 16. Composite functions Functions & Graphs 16 푓 ∘ 푔 푥 = 푓 푔 푥 Inner function Outer function ℛ푔 ⊂ 퐷푓 Is required! If not, 풟푔 must be restricted 퐷푔 ℛ푔 퐷푓 ℛ푓 퐷푓∘푔 ℛ푓∘푔 Domain restriction
- 17. Identity function 푓: 푥 ⟼ 푥 풟 = ℝ ℛ = ℝ Functions & Graphs 17
- 18. Inverse function 푓−1 Is the inverse function of 푓 if: 푓 ∘ 푓−1 푥 = 푓−1 ∘ 푓 푥 = 푥 퐷푓−1 = ℛ푓 푅푓−1 = 퐷푓 Only one-to-one functions are invertible ! Functions & Graphs 18
- 19. END Disclaimer This document is meant to be apprehended through professional teacher mediation (‘live in class’) together with a mathematics text book, preferably on IB level. Functions & Graphs 19