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1.3 Twelve Basic Functions

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  1. 1.3 Twelve Basic Functions After today’s lesson you will be able to: • Recognize graphs of twelve basic functions and describe their characteristics • Determine domains of functions related to the twelve basic functions • Combine the twelve basic functions in various ways to create new functions

  2. The Identity Function • f(x) = x • Domain: (- ∞, ∞) • Range: (- ∞, ∞) • Unbounded • Always increasing

  3. The Squaring Function • f(x) = x2 • Domain: (-∞, ∞) • Range: [0, ∞) • Bounded below • Decreasing: (-∞, 0) • Increasing: (0,+∞) • Graph called parabola. How can you use the squaring function to describe g(x) = x2 – 2?

  4. The Cubing Function • f(x) = x3 • Domain: (- ∞, ∞) • Range: (- ∞, ∞) • Unbounded • Always increasing • Note: The origin is called a point of inflection since the graph changes curvature at this point.

  5. The Reciprocal Function • f(x) = 1/x • Domain: (- ∞,0) U (0, ∞) • Range: (- ∞,0) U (0, ∞) • Vertical asymptote: x = 0 (y-axis) • Horizontal asymptote: y = 0 (x-axis) • Note: Graph called a hyperbola

  6. The Square Root Function • Domain: [0, ∞) • Range: [0, ∞) • Bounded below (0,0). • Graph is top-half of a sideways parabola.

  7. The Exponential Function • f(x) = ex • Domain: (- ∞, ∞) • Range: (0, ∞) • Bounded below by y = 0 (x-axis) • Value of e ≈2.718 • Always increasing • Graph is half of a hyperbola. Use the exponential function to describe g(x) = -ex.

  8. The Natural Logarithm Function • f(x) = ln x • Domain: (0, ∞) • Range: (- ∞, ∞) • Bounded on the left by x = 0 (y-axis) • Note: Inverse of exponential function. • Used to describe many real-life phenomena included intensity of earthquakes (Richter scale).

  9. The Sine Function • f(x) = sin x • Domain: (- ∞, ∞) • Range: [-1,1] • Note: Periodic function (repeats every 2π units). • Local Maximum: π/2 + kp, where k is an odd integer • Local Minimum: kp, where k is an integer • Bounded Use the graph of the sine function to describe the graph of g(x) = 2 sin x.

  10. The Cosine Function • f(x) = cos x • Domain: (- ∞, ∞) • Range: [-1,1] • Note: Periodic function (repeats every 2p units). • Local Maximum: kp, where k is an even integer • Local Minimum: kp/2, where k is an odd integer • Bounded Use the graph of the cosine function to describe the graph of g(x) = - ½ cos x.

  11. The Absolute Value Function • f(x) = |x| • Domain: (- ∞, ∞) • Range: [0, ∞) • Bounded below at the origin • V-shaped graph Use the graph of the absolute value function to describe the graph of g(x) = |x | + 2 .

  12. The Greatest Integer Function • f(x) = [x] = int (x) • Domain: (- ∞, ∞) • Range: All integers • Continuous at each non-integer value • Jump discontinuity at each integer • Also called the “step” function.

  13. The Logistic Function • f(x) = • Domain: (- ∞, ∞) • Range: (0,1) • Bounded • Note: Model for many applications of biology (population growth) and business.

  14. Looking for Domains • Nine of the functions have domain the set of all real numbers. Which 3 do not? • One of the functions has domain the set of all reals except 0. Which function is it, and why is 0 not in the domain? • Which of the two functions have no negative numbers in their domains? Of these two, which one is defined at 0?

  15. Looking for continuity • Only two of the twelve functions have points of discontinuity. Which functions are they? Are these points in the domain of the function?

  16. Looking for boundedness • Only three of the twelve basic functions are bounded (above and below). Which 3?

  17. Looking for symmetry • Three of the twelve basic functions are even. Which are they?

  18. Which of the basic functions are continuous? For the functions that are discontinuous, identify as infinite or jump.

  19. Which of the basic functions have symmetry? Describe each.

  20. Exploration: Looking for Asymptotes • Two of the basic functions have vertical asymptotes at x = 0. Which two? • Form a new function by adding these functions together. Does the new function have a vertical asymptote at x = 0? • Three of the basic functions have horizontal asymptotes at y = 0. Which three? • Form a new function by adding these functions together. Does the new function have a horizontal asymptote y = 0? • Graph f(x) = 1/x, g(x) = 1/(2x2-x), and h(x) = f(x) + g(x). Does h(x) have a vertical asymptote at x = 0? Explain.

  21. Piecewise-defined Functions Which of the twelve basic functions has the following piecewise definition over separate intervals of its domain? x if x ≥ 0 f(x) = -x if x < 0

  22. Piece-Wise Functions

  23. Use the basic functions from this lesson to construct a piecewise definition for the function shown.Is your function continuous?

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