Topic 1 : Elementary functions

1
Topic 1

• Topic 1 Elementary functions
• Reading Jacques
• Section 1.1 - Graphs of linear equations
• Section 2.1 Quadratic functions
• Section 2.2 Revenue, Cost and Profit

2
Linear Functions

• The function f is a rule that assigns an incoming
  number x, a uniquely defined outgoing number y.
• y f(x)
• The Variable x takes on different values...
• The function f maps out how different values of x
  affect the outgoing number y.
• A Constant remains fixed when we study a
  relationship between the incoming and outgoing
  variables

3
Simplest Linear Relationship

• y abx ? independent
• dependent ? ? variable
• variable intercept
• This represents a straight line on a graph i.e.
  a linear function has a constant slope
• b slope of the line change in the dependent
  variable y, given a change in the independent
  variable x.
• Slope of a line ?y / ?x
• (y2-y1) / (x2-x1)

4
Example Student grades

• Example
• y a bx
• y is the final grade,
• x is number of hours studied,
• a guaranteed

• Consider the function
• y 5 0x
• What does this tell us?
• Assume different values of x

5
Example Continued What grade if you study 0
hours? 5 hours?

• y50x
• Output constant slope Input
• y a b X
• 5 5 0 0
• 5 5 0 1
• 5 5 0 2
• 5 5 0 3
• 5 5 0 4
• 5 5 0 5

6
Example Continued.

• y515x

If x 4, what grade will you get? Y 5 (4
15) 65
7
Demand functions The relationship between price
and quantity
If p 5, how much will be demanded? D 10 - (2
5) 0
8
Inverse Functions

• Definition
• If y f(x)
• then x g(y)
• f and g are inverse functions
• Example
• Let y 5 15x
• If y is 80, how many hours per week did they
  study?

9
Example continued..

• If y is 80, how many hours per week did they
  study?
• Express x as a function of y 15x y 5....
• So the Inverse Function is x (y-5)/15
• Solving for value of y 80
• x (80-5 / 15)
• x 5 hours per week

10
An inverse demand function

• If D a bP then the inverse demand curve is
  given by P (a/b) (1/b)D
• E.g. to find the inverse demand curve of the
  function D 10 -2P
• First, re-write P as a function of D
• 2P 10 D
• Then, simplify
• So P 5 0.5D is the inverse function

11
More Variables

• Student grades again
• y a bx cz
• y is the final grade,
• x is number of hours studied,
• z number of questions completed
• a guaranteed

• Example
• If y 5 15x 3z, and a student studies 4 hours
  per week and completes 5 questions per week, what
  is the final grade?
• Answer
• y 5 15x 3z
• y 5 (154) (35)
• y 56015 80

12
Another example Guinness Demand.

• The demand for a pint of Guinness in the Student
  bar on a Friday evening is a linear function of
  price. When the price per pint is 2, the demand
  is 6 pints. When the price is 3, the demand is
  only 4 pints. Find the function
• D a bP

• 6 a 2b gt a 6-2b
• 4 a 3b gt a 4-3b
• 6-2b 4-3b
• Solving we find that b -2
• If b -2, then a 6-(-4) 10
• The function is D 10 2P
• What does this tell us??
• Note, the inverse Function is
• P 5- 0.5D

13
A Tax Example.

• Answer
• THP E 0.4 (E 4000) if Egt4000
• THP E 0.4 (4000-E)
• if Elt4000
• In both cases,
• THP 1600 0.6E So
• i) If E 4000 gt
• THP 160024004000
• ii) If E 5000 gt
• THP 160030004600
• iii) If E 3000 gt
• THP 160018003400

• let 4000 be set as the target income. All income
  above the target is taxed at 40. For every 1
  below the target, the worker gets a negative
  income tax (subsidy) of 40.
• Write out the linear function between take-home
  pay and earnings.

14
Tax example continued.

• THP 1600 0.6E
• If the hourly wage rate is equal to 3 per hour,
  rewrite take home pay in terms of number of hours
  worked?
• Total Earnings E (no. hours worked X hourly
  wage)
• THP 1600 0.6(3H) 1600 1.8H
• Now add a (tax free) family allowance of 100
  per child to the function THP 1600 0.6E
• THP 1600 0.6E 100Z (where z is number of
  children)
• Now assume that all earners are given a 100
  supplement that is not taxable,
• THP 1600 0.6E 100Z 100
• 1700 0.6E 100Z

15
Topic 1 continued Non- linear
EquationsJacques Text Book Sections 2.1 and
2.2
16
Quadratic Functions

• Represent Non-Linear Relationships
• y ax2bxc where a?0, cIntercept
• a, b and c are constants
• So the graph is U-Shaped if agt0,
• And Hill-Shaped if alt0
• And a Linear Function if a0

17
Solving Quadratic Equations

• 1) Graphical Approach To find Value(s), if any,
  of x when y0, plot the function and see where it
  cuts the x-axis
• If the curve cuts the x-axis in 2 places there
  are always TWO values of x that yield the same
  value of y when y0
• If it cuts x-axis only once when y0 there is a
  unique value of x
• If it never cuts the x-axis when y0 there is no
  solution for x

18
e.g. y -x24x5
Since alt0 gt Hill Shaped Graph
19
The graph
y0, then x 5 OR x -1
20
Special Case a1, b0 and c0So y ax2bxc
gt y x2
Min. Point (0,0)
Intercept 0
21
Practice examples

• Plot the graphs for the following functions and
  note (i) the intercept value (ii) the value(s),
  if any, where the quadratic function cuts the
  x-axis
• y x2-4x4
• y 3x2-5x6

22
Solving Quadratic Equations

• 2) Algebraic Approach find the value(s), if any,
  of x when y0 by applying a simple formula

23
Example

• e.g. y -x24x5
• hence, a -1 b4 c5
• Hence, x 5 or x -1 when y0
• Function cuts x-axis at 5 and 1

24
Example 2

• y x2-4x4
• hence, a 1 b - 4 c4
• If y 0

Function only cuts x-axis at one point, where x2

x 2 when y 0
25
Example 3

• y 3x2-5x6
• hence, a 3
• b - 5 c6
• If y 0

when y 0 there is no solution
The quadratic function does not intersect the
x-axis
26
Understanding Quadratic Functions

• intercept where x0 is c
• agt0 then graph is U-shaped
• alt0 then graph is inverse-U
• a 0 then graph is linear
• b2 4ac gt 0 cuts x-axis twice
• b2 4ac 0 cuts x-axis once
• b2 4ac lt 0 no solution

27
Essential equations for Economic Examples

• Total Costs TC FC VC
• Total Revenue TR P Q
• ? Profit TR TC
• Break even ? 0, or TR TC
• Marginal Revenue MR change in total revenue
  from a unit increase in output Q
• Marginal Cost MC change in total cost from a
  unit increase in output Q
• Profit Maximisation MR MC

28
An Applied Problem

• A firm has MC 3Q2- 32Q96
• And MR 236 16Q
• What is the profit Maximising Output?

• Solution
• Maximise profit where MR MC
• 3Q2 32Q 96 236 16Q
• 3Q2 32Q16Q 96 236 0
• 3Q2 16Q 140 0
• Solve the quadratic using the formula
• where a 3 b -16 and c -140
• Solution
• Q 10 or Q -4.67
• Profit maximising output is 10 (negative Q
  inadmissable)

29
Graphically
30
Another Example.

• If fixed costs are 10 and variable costs per unit
  
• are 2, then given the inverse demand function P
  14 2Q
• Obtain an expression for the profit function in
  terms of Q
• Determine the values of Q for which the firm
  breaks even.
• Sketch the graph of the profit function against Q

31
Solution

• Profit TR TC P.Q (FC VC)
• ? (14 - 2Q)Q (2Q 10)
• ? -2Q2 12Q 10
• Breakeven where Profit 0
• Apply formula to solve quadratic where ? 0
• so solve -2Q2 12Q 10 0 with
• Solution at Q 1 or Q 5 the firm breaks even

32
3. Graphing Profit Function

• STEP 1 coefficient on the squared term
  determines the shape of the curve
• STEP 2 constant term determines where the graph
  crosses the vertical axis
• STEP 3 Solution where ? 0 is where the graph
  crosses the horizontal axis

33
(No Transcript)
34
Questions Covered on Topic 1 Elementary
Functions

• Linear Functions and Tax
• Finding linear Demand functions
• Plotting various types of functions
• Solving Quadratic Equations
• Solving Simultaneous Linear (more in next
  lecture)
• Solving quadratic functions