ECON 1150, 2013 Functions of One Variable

ECON 1150, Functions of One Variable Examples: y = 1 + 2x, y = x 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 + x 2

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

ECON 1150, 2013 Example 1.2: Let f(x) = (x 2 – 1) / (x 2 + 1). a.Find f(b/a). b.Find f(b/a) + f(a/b). c.f[ f(b/a) ].

ECON 1150, 2013 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 The graph of a function: The set of all points (x, f(x)). Example 1.4: a.Find some of the points on the graph of g(x) = 2x – 1 and sketch it. b.Consider the function f(x) = x 2 – 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 Example 1.5: Determine the domain and range of the function

ECON 1150, 2013

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

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

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

ECON 1150, 2013 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 Linear functions: Constant slope Non-linear functions: Variable slope y = x y = x 2 y = 6 + x 0.5

ECON 1150, Polynomials 3 4 = 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 a n = a  a  …  a (n times) base: a exponent: n

ECON 1150, 2013 a n ·a m = a n+m, a n /a m = a n-m, (a n ) m = a nm, (a·b) n = a n ·b n, (a/b) n = a n /b n, a -n = 1 / a n a 0 = 1 General properties of exponents For any real numbers a, b, m and n,

ECON 1150, 2013 Power function: y = f(x) = ax b, a  0 Example 1.9: If ab 2 = 2, compute the following: a. a 2 b 4 ; b. a -4 b -8 ; c. a 3 b 6 + a -1 b -2. Example 1.10: Sketch the graphs of the function y = x b for b = -1.3, 0.3, 1.3.

ECON 1150, 2013 Linear functions: y = a + bx Quadratic functions y = ax 2 + 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 = x 2 + x + 1; (b) y = -x 2 + x + 2.

ECON 1150, 2013 Cubic functions y = ax 3 + bx 2 + 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: (a) y = -x 3 + 4x 2 – x – 6; (b) y = 0.5x 3 – 4x 2 + 2x + 2.

ECON 1150, 2013 Polynomial of degree n y = a n x n a 2 x 2 + a 1 x + a 0 where n is any non-negative integer and a n  0. n = 1: Linear function n = 2: Quadratic function n = 3: Cubic function

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

ECON 1150, 2013 Exponential function: y = Ab t

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

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

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

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

ECON 1150, 2013 Logarithmic and Exponential Functions

ECON 1150, 2013 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 x = 10; c. e x + 4e -x = 4.