Functions and Graphs

• Function : is represented as f(x)=y such as y = x2

• an equation gives ‘ y as a function of x’ means that for every x value there is unique y value.

• Domain: the domain of the function consists of all possible x values.

• Range: the range of a function consists of all possible y values.

• Graph of function of y=x2 • Domain: - x +• Range y > 0;

Even and Odd Functions

• Even function: the function is even if its graph is symmetric with respect to the y axis.

• If f(-x) = f(x)

• Odd function: the function is odd if its graph is symmetric with respect to the origin.

• Graph of function f =x3 Its odd because f(-x)= -x3 = -f(x)And it is symmetric to the origin.

Combination of functions and inverse functions

• Combination of functions: if f(x) and g(x) two functions then f(x) + g(x), f(x) – g(x) and f(x)/g(x) are combination of two functions.

• Domain: the domain is the intersection of the domain of f(x) and g(x). In other words their domain is where the domain of f(x) overlaps the domain of the g(x).

• Note that for f(x)/g(x) if g(x)0.

Examples

• Find f(x) + g(x), f(x) - g(x), and f(x)/g(x) and their domain if f(x)= x2 -4 and g(x) = x+2.

• Solution:• f(x) + g(x) = x2 -4 + x+2 = x2 +x -2• f(x) - g(x) = x2 -4 – x -2 = x2 - x -6• f(x) / g(x) = = (x2 -4 )/ (x+2)= (x -2)

• Graph of function f(x) = = x2 -4 and g(x) = x+2 domain is - x +

Example

• Find f g(x), g f(x), f g(-1), f g(0) and g f(1) for f(x)= x2 – 1 and g(x)= x+1

Inverse functions

• The inverse of a function f(x) is f-1(x).• We use functional decomposition to show that

two functions are inverse of each other.• F(x) is inverse of g(x) if f g(x)= x and

g f(x) = x.

• If the horizontal line touches the graph more than one place then the function will not have inverse.

• Find the inverse function to the following– Y=x2+4

Exponential functions

• The function of the form f(x) = ax , where a positive number we call It exponential function.

• Draw the graph of F(x)= 2x

Logarithmic functions

• In logarithmic functions y= loga x • Where the exponential function (inverse of

the logarithmic function) x=ay • Draw the graph of f(x)= log2 x;

• Rewrite the logarithmic function as exponential function– Log100 10 = ½

– Log(x+1) 9=2

– Log7 1/49=-2

– Loge 2 = 06931

• Solutions– 10=100 ½ – 9 = (x+1)2

– 1/49= 7-2 – 2 = e 0.6931

• Rewrite the exponential function as logarithmic functions.

• 125 1/3 = 5• 10 -4 = 0.0001• e-½ = 1.6484• 8x = 5

• Solution• Log125 5 = 1/3

• Log10 0.0001=-4

• Loge 1.6487 =1/2

• Log8 5 = x

• Solve the following exercises– x+y11 ; y x find the point having whole number

coordinates and satisfy these inequalities which gives • The maximum value of x + 4y• The minimum value of 3x+y

• 3x + 2y > 24 ; x+y <12; y<1/2 x; y >1.• Find the point having whole number

coordinates and satisfying these inequalities which gives – The maximum value of 2x +3y– The minimum value of x + y

• 3x + 2y 60; x+2y 30; x >10; y>0.• Find the point having whole number

coordinates and satisfying these inequalities which gives – The maximum value of 2x+y– The minimum value of xy.