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Functions of One Variable

Functions of One Variable. 1 . Functions of One Variable. Examples: y = 1 + 2x, y = -2 + 3x. Let x and y be 2 variables. When a unique value of y is determined by each value of x, this relation is called a function. General form of function: y = f(x) read “y is a function of x.”

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Functions of One Variable

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  1. Functions of One Variable ECON 1150, 2013

  2. 1. Functions of One Variable Examples: y = 1 + 2x, y = -2 + 3x Let x and y be 2 variables. When a unique value of y is determined by each value of x, this relation is called a function. General form of function: y = f(x) read “y is a function of x.” y: Dependent variable x: Independent variable Specific forms: y = 2 + 5x y = 80 + x2 ECON 1150, 2013

  3. Example 1.1: • Let f(x) = a + bx. Given that f(0) = 2 and f(10) = 32. Find this function. • Let f(x) = x² + ax + b and f(-3) = f(2) = 0. Find this function and then compute f( + 1). ECON 1150, 2013

  4. Example 1.2: Let f(x) = (x2 – 1) / (x2 + 1). • Find f(b/a). • Find f(b/a) + f(a/b). • f[ f(b/a) ]. ECON 1150, 2013

  5. Domain of a function: The possible values of the independent variable x. Range of a function: The values of the dependent variables corresponding to the values of the independent variable. Example 1.3: y  0 0  y  1 ECON 1150, 2013

  6. The graph of a function: The set of all points (x, f(x)). • Example 1.4: • Find some of the points on the graph of g(x) = 2x – 1 and sketch it. • Consider the function f(x) = x2 – 4x + 3. Find the values of f(x) for x = 0, 1, 2, 3, and 4. Plot these points in a xy-plane and draw a smooth curve through these points. ECON 1150, 2013

  7. Example 1.5: Determine the domain and range of the function ECON 1150, 2013

  8. ECON 1150, 2013

  9. y y y = ax + b (a < 0) y = ax + b (a > 0) b a a 1 1 x 0 b x 0 Positive slope (a > 0) Negative slope (a < 0) 1.1 Linear Functions General form of linear functions y = ax + b (a and b are called parameters.) Intercept: b Slope: a ECON 1150, 2013

  10. The slope of a linear function = a y-intercept y2 – y1y = - ------------------ = ------------ = ------ x-intercept x2 – x1x • Example 1.6: • Find the equation of the line through (-2, 3) with slope -4. Then find the y-intercept and x-intercept. • Find the equation of the line passing through (-1,3) and (5,-2). ECON 1150, 2013

  11. Example 1.7: a. Keynesian consumption function: C = 200 + 0.6Y Intercept = autonomous consumption = 200 Slope = MPC = 0.6 b. Demand function: Q = 600 – 6P This function satisfies the law of demand. ECON 1150, 2013

  12. Example 1.8: Assume that consumption C depends on income Y according to the function C = a + bY, where a and b are parameters. If C is $60 when Y is $40 and C is $90 when Y is $80, what are the values of the parameters a and b? ECON 1150, 2013

  13. y = x2 y = 5 + 0.2x y = 6 + x0.5 Linear functions: Constant slope Non-linear functions: Variable slope ECON 1150, 2013

  14. 1.2 Polynomials 34 = 3  3  3  3 = 81 (-10)3 = (-10)  (-10)  (-10) = - 1,000 If a is any number and n is any natural number, then the nth power of a is an = a  a  …  a (n times) base: a exponent: n ECON 1150, 2013

  15. General properties of exponents For any real numbers a, b, m and n, • an·am = an+m, • an/am = an-m, • (an)m = anm, • (a·b)n = an·bn, • (a/b)n = an/bn, • a-n = 1 / an • a0 = 1 ECON 1150, 2013

  16. Example 1.9: If ab2 = 2, compute the following: • a2b4; • a-4b-8; • a3b6 + a-1b-2. Power function: y = f(x) = axb, a  0 Example 1.10: Sketch the graphs of the function y = xb for b = -1.3, 0.3, 1.3. ECON 1150, 2013

  17. Linear functions: y = a + bx Quadratic functions y = ax2 + bx + c (a  0) a > 0  The curve is U-shaped a < 0  The curve is inverted U-shaped Example 1.11: Sketch the graphs of the following quadratic functions: (a) y = x2 + x + 1; (b) y = -x2 + x + 2. ECON 1150, 2013

  18. Cubic functions y = ax3 + bx2 + cx +d (a  0) a > 0: The curve is inverted S-shaped. a < 0: The curve is S-shaped. • Example 1.12: Sketch the graphs of the cubic functions: • y = -x3 + 4x2 – x – 6; • y = 0.5x3 – 4x2 + 2x + 2. ECON 1150, 2013

  19. Polynomial of degree n y = anxn + ... + a2x2 + a1x + a0 where n is any non-negative integer and an 0. n = 1: Linear function n = 2: Quadratic function n = 3: Cubic function ECON 1150, 2013

  20. 1.3 Other Special Functions Exponential function: y = Abt, b > 1 t: Exponent a: Base The exponent is a variable. Example 1.13: Let y = f(t) = 2t. Then f(3) = 23 = 8 f(-3) = 2-3 = 1/8 f(0) = 20 = 1 f(10) = 210 = 1,024 f(t + h) = 2t+h ECON 1150, 2013

  21. Exponential function: y = Abt ECON 1150, 2013

  22. The Natural Exponential Function f(t) = Aet. Examples of natural exponential functions: y = et; y = e3t; y = Aert or y = exp(t); y = exp(3t); y = Aexp(rt). ECON 1150, 2013

  23. y = e-x y = ex Two Graphs of Natural Exponential Functions ECON 1150, 2013

  24. Example 1.14: Which of the following equations do not define exponential functions of x? a. y = 3x; b. y = x2; c. y = (2)x; d. y = xx; e. y = 1 / 2x. ECON 1150, 2013

  25. Natural logarithm y = logex = lnx Logarithmic function y = bt t = logby We say that t is the logarithm of t to the base of b. Rules of logarithm ln(ab) = lna + lnb ln(a/b) = lna – lnb ln(xa) = alnx x = elnx ln(1) = 0 ln(e) = 1 lnex = x ECON 1150, 2013

  26. Logarithmic and Exponential Functions ECON 1150, 2013

  27. Example 1.15: Find the value of f(x) = ln(x) for x = 1, 1/e, 4 and -6. Example 1.16: Express the following items in terms of ln2. a. ln4; b. ln(3(32)); c. ln(1/16). Example 1.17: Solve the following equations for x: a. 5e-3x = 16; b. 1.08x = 10; c. ex + 4e-x = 4. ECON 1150, 2013

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