577/6553 Mathematical Methods / General Mathematics G 2014/s1

Lecture 2Functions 2

Dr Peter Vassiliou

Week 2

MMeths/GMG 2014/s1, Lecture 2 p.1

The straight line - (sec. 1.1)In the USA they use the Fahrenheit temperature scale and inAustralia we use Celsius. What is the relationship?

Its a “straight line relationship”!

The formula expressing degrees Fahrenheit (F) as a function ofdegrees Celsius (C) is

F(C) =95

C+32.

Graph looks like:

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C0 20 40 60 80 100

F

0

50

100

150

200

F as a function of C

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The straight lineIn this example and in others the relationship is given by alinear model. i.e. F(C) = mC+b. More generally f (x) = mx+bor y = mx+b. The graph of such a function is always a straightline.

Number m is said to be the slope or gradient of the line.Number b is the y-intercept.

Example: f (x) = 2x−1

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The equation of a lineThe slope of a straight line passing through the points A(x1,y1)& B(x2,y2) is “vertical rise over horizontal run”

m =y2− y1

x2− x1

Example. What is the equation of the straight line containingpoints A(−1,2) and B(2,3)?

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Exercises with lines

Find the equation of the line with slope 2 and that goesthrough the point (1,3).

Find the equation of the line containing points A(2,3) andB(4,7).

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Parallel lines

The lines y = mx+b and y = nx+a are parallel if and only ifm = n.

Find a line parallel to y = 3x+2 that passes through thepoint (2,2).

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The lines y = mx+b and y = nx+a are perpendicular if andonly if mn =−1.

Find the line perpendicular to y = 2x+3 that passesthrough the point (1,2).

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InterceptsThe x-intercepts are the points where the graph crosses the xaxis.

Given a function, how can we find the x- intercepts?

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The y-intercepts are the points where the graph crosses the yaxis.

If we have a function, how can we find the y-intercepts?

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Exponential Function - (sec. 2.1)An exponential function is a function of the form: y = f (x) = bx

where b > 0 and b 6= 1.

Are the following exponential funtions?3x, 5x, x2

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Graph of an exponential functionWhat is the domain and range of an exponential function?If b > 1: for example, f (x) = 2x

If 0 < b < 1: for example, g(x) = (12)

x

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Exponential function rulesLet b > 0. Then for all x,y

1 bx.by = bx+y

2 bx

by = bx−y

3 (bx)y = bx.y

4 (b.c)x.= bx.cx

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(bc )

x = bx

cx

b1 = b

b0 = 1

b−x = 1bx = (1

b)x

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One-to-One FunctionsWe say that a function is one-to-one if and only if for eachdifferent input gives different outputs.

Example f (x) = x3 is one-to-one

Example f (x) = x2 is not one-to-one if the domain of f is−a < x < a for any a > 0.

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Horizontal Line TestIf we have a graph of a function, the horizontal line test can tellus if the function is one-to-one.

Horizontal Line Test(a) If there is a line parallel to the x-axis (a horizontal line),that cuts the graph at least in two points, then the functionis not one-to-one.(b) If each horizontal line cuts at most in one point of thegraph, then the function is one-to-one.

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Inverse functionsGiven a function f (x) where for each input we get an output, theinverse function of f is a function f−1, that for each output (fromf ) gives us an input from f .

Example. Degrees Fahrenheit (F) as a function of degreesCelcius (C): F = 9

5C+32. What if we know the degreesFahrenheit and we want degrees Celsius? Then we computethe inverse of the function C 7→ 9

5C+32.

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Example. What is f−1 if f (x) = 95 x+32?

Example. What is f−1 if f (x) = 1x −1?

Do all the functions have an inverse?

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Theorem: A function has an inverse if and only if the function isone-to-one.This means

If a function has an inverse then it is one-to-one.If a function is one-to-one then it has an inverse.

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If we have a function that is not one-to-one (therefore it doesn’thave an inverse), we can reduce its domain enough, so thefunction becomes one-to-one.

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Graph of f−1

If we have the graph of f (x) we can graph f−1(x). The graph off−1(x) is the reflexion of f (x) respect the line y = x.

Definition. The range of a function is the set of all its outputs.

range of f = domain of f−1

domain of f = range of f−1

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Examples.

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Logarithmic functions - (sec. 2.2)The inverse of the exponential function is the logarithm function.

Definition: y = loga x if and only if x = ay.

ie. y = loga x is the inverse function of y = ax and the functiony = ax is the inverse of y = loga x.

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Examples:

1 log2 x = 4, what is x?

2 loge(x+1) = 7, what is x?

3 log3(127) = y, what is y?

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How to graph the logarithm function?What is its domain and range? If a > 1, for example, y = log2 x

If 0 < a < 1, for example, y = log 12

x

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