ECON 1150, 2013

Functions of One Variable

ECON 1150, 2013

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

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) = (x2 – 1) / (x2 + 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:

1y1- :range 1x1- :domain ,x-1 y d.

numbers real all :range 1 x:domain ,x-1 y c.

0 y :range numbers real all domain , xy .b

numbers real all range numbers real all domain ,x21y .a

2

2

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) = 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

Example 1.5: Determine the domain and range of the function

.2x1x

1y

ECON 1150, 2013

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

.2x1x

1y

ECON 1150, 2013

General form of linear functions

y = ax + b (a and b are called parameters.)

Intercept: b Slope: a

bx

y

0

a1

y = ax + b (a > 0)

Positive slope (a > 0)

b

0x

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 y2 – y1 y= - ------------------ = ------------ = ------ x-intercept x2 – x1 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 = 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

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

52.50-2.5-5

6

5.5

5

4.5

4

x

y

x

y

y = 5 + 0.2x

52.50-2.5-5

25

20

15

10

5

0

x

y

x

y

y = x2

53.752.51.250

8

7.5

7

6.5

6

x

y

x

y

y = 6 + x0.5

ECON 1150, 2013

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

• 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

General properties of exponents

For any real numbers a, b, m and n,

ECON 1150, 2013

Power function: y = f(x) = axb, a 0

Example 1.9: If ab2 = 2, compute the following:

a. a2b4;

b. a-4b-8;

c. a3b6 + a-1b-2.

Example 1.10: Sketch the graphs of the function y = xb for b = -1.3, 0.3, 1.3.

ECON 1150, 2013

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

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:

(a) y = -x3 + 4x2 – x – 6;

(b) y = 0.5x3 – 4x2 + 2x + 2.

ECON 1150, 2013

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

1.3 Other Special Functions

t: Exponent a: BaseThe exponent is a variable.

Exponential function: y = Abt, b > 1

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

Exponential function: y = Abt

x

y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

1

2

3

4

y = 2^x

y = 2^0.5

ECON 1150, 2013

en

n

n

11lim718281828.2 base Natural

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 = Aexp(rt).

ECON 1150, 2013

Two Graphs of Natural Exponential Functions

x

y

-2 -1 0 1 2

1

2

3

4

y = exy = e-x

ECON 1150, 2013

Example 1.14: Which of the following equations do not define exponential functions of x?

a. y = 3x; b. y = x2; c. y = (2)x;

d. y = xx; e. y = 1 / 2x.

ECON 1150, 2013

Logarithmic function

y = bt t = logby

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

Natural logarithm y = logex = lnx

We say that t is the logarithm of t to the base of b.

ECON 1150, 2013

x

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

1

2

3

4y = e^x

y = lnx

y = x

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. 1.08x = 10; c. ex + 4e-x = 4.