Functions PDF

Function A function is a rule that assigns to each element in a set D exactly one element, called f(x) , in a set E. Domain of f(x) is the set of all the values of x that will make f(x) have real values The range of is the set of all possible values of f(x) as varies throughout the domain. x is ind...

Function A function is a rule that assigns to each element in a set D exactly one element, called f(x) , in a set E. Domain of f(x) is the set of all the values of x that will make f(x) have real values The range of is the set of all possible values of f(x) as varies throughout the domain. x is independent variable And y = f(x) is the dependent variable the vertical line test The vertical line test A curve in the xy-plane is the graph of some function f if and only if no vertical line intersects the curve more than once. This curve cannot be the graph of y as a function of x. no vertical line intersects the curve more than once. This curve is the graph of a function Example The graph of the equation x2 + y2 = 25 Sol is a circle of radius 5 centered at the origin and hence there are vertical lines that cut the graph more than once. This equation does not define y as a function of x Example In each part of the accompanying figure, determine whether the graph defines y as a function of x. Yes Yes No No EXAMPLE 1 The graph of f(x) a function is shown in Figure. (a) Find the values of f(1) and f(5). (b) What are the domain and range of f(x) ? Sol a) f (1) =3 and f(5)= -0.7 from the figure b) The domain of is the closed interval [0 , 7]. Range is Example 2 Sketch the graph and find the domain and range of each function. The domain of is the set of all real numbers R=(−∞, ∞). The range is also R =(−∞, ∞). The domain of is the set of all real numbers R The range is =[0, ∞). Elementary Functions Examples is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic polynomial. is a polynomial of degree 7, as 7 is the highest power of x. We need to know only the highest power of x to find out the degree. Which is a polynomial of degree 2, as 2 is the highest power of x. This is called a quadratic. Functions containing other operations, such as square roots, are not polynomials. is not a polynomial as it contains a square root. is not a polynomial as it contains a ‘divide by x’. a) Constant function f(x) = c = constant Domain is R (all real values) Range is {c} Example find the domain and the range of f(x) =2 Solution Domain is R (all real values) Range is {2} Example find the domain and the range of f(x) =-10 Solution Domain is R (all real values) Range is {-10} Example find the domain and the range of f(x) =20 Solution Domain is R (all real values) Range is {20} b) linear function f(x)=a x +c Domain is R (all real values) Range is R (all real values) Example find the domain and the range of f(x) =2x + 1 Solution Domain is R (all real values) Range is R (all real values) Example find the domain and the range of f(x) =5x + 10 Solution Domain is R (all real values) Range is R (all real values) Example find the domain and the range of f(x) =-4x + 13 Solution Domain is R (all real values) Range is R (all real values) c) Quadratic functions f(x) = a0 +a1 x+ a2 x2 Domain is R (all real values) Range is from the vertex to ∞ 𝟎𝒓 − ∞ Here are some examples of quadratic functions:  Domain is R and the Range is = [0, ∞) Domain is R and the Range is = (- ∞, 0] To get the vertex of the quadratic functions If n=2: y=a0+a1 x+ a2 x2 Vertex of the parabola : V(α=-a1/2 a2 , β=f (α ) ) Domain of f(x) = R The range is [β, ∞) or (−∞ , 𝛃 ] Where β is the max or min value of y Example A parabola, which has vertex (3, -3), is sketched below. Find the domain and range of this function. Domain is R (all real values). The range is all real y ≥ -3 =[−𝟑, ∞) Example find the domain and range of f(x) = 3x - x2 Solution To get the vertex y=a0+a1 x+ a2 x2 a1 = 3 & a2= -1 Vertex of the parabola : V(α=-a1/2 a2 , β=f (α ) ) 𝐚𝟏 −𝟏 ∗ 𝟑 𝟑 𝛂=− = = 𝟐𝐚𝟐 𝟐 ∗ (−𝟏) 𝟐 𝟑 𝟑 𝟐 𝛃 = 𝐟 (𝛂 ) = 𝟑 ∗ − ( ) = 𝟐. 𝟐𝟓 𝟐 𝟐 Domain is R (all real values). Range is all real y where y ≤ 2.25 = =(−∞, 𝟐. 𝟐𝟓] Example find the domain and range of f(x) = (x - 1)2 + 1 Sol f(x) = (x - 1)2 + 1=x2-2x+2 If n=2: y=a0+a1 x+ a2 x2 Vertex of the parabola : V(α=-a1/2 a2 , β=f (α ) ) 𝐚𝟏 −𝟏 ∗ −𝟐 𝛂=− = =𝟏 𝟐𝐚𝟐 𝟐 ∗ (𝟏) 𝛃 = 𝐟 (𝛂 ) = (𝟏)𝟐 − 𝟐 ∗ 𝟏 + 𝟐 = 𝟏 So the vertex at (1,1) Domain is R (all real values). Range is all real y where y ≥ 1 =[𝟏, ∞) If n is odd like x3 , x5 , x9 , x15 , x19 , x33 Domain of f(x) = R Range of f(x) = R Example Find the domain and range of the following functions. 1) f(x) =x3+2x2 -3x+4 The domain = R & Range = R 2) f(x) =x7+2x5 +2x4 -3x+5 The domain = R & Range = R 3) f(x) =x5+2x4 -3x+7 The domain = R & Range = R