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 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+y11 ; 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.