Differential Calculus Engr. Joeydann M. Telin Chapter 1 FUNCTIONS Definition: - A function from set X and Y is the set of ordered pairs of distinct real numbers (x,y), in which no two distinct ordered pairs have the same first component. The set of permissible values of x is called the domain of the function and the set of all resulting value of y is called the range of the function. The symbol f(x) (read as “f of x”) denotes the particular value of y that corresponds to the given value of x. The variable x is called the independent variable while the variable y is called the dependent variable because y is usually expressed in terms of x when their relationship is given in the form of an equation. In our high school days Functions can be express by: • Arrow diagrams • Tables • Graphs • Equations y = 𝑥2 Functions can be specified in a variety of ways. Functions that are given by equations involving the dependent and independent variables. For instance, the equation Equation in implicit form defines the dependent variable, as a function of the independent variable. To evaluate this function (that is, to find the value that corresponds to a given -value), it is convenient to isolate on the left side of the equation. Equation in explicit form Using as the name of the function, you can write this equation as Function notation Ex 1. Given that f is the function defined by: f(x) = 𝑥 2 + 3𝑥 − 4 Find: a.) f(0) e.) f(3x) b.) f(3) f.) f(x+h) c.) f(h) g.) f(x) + f(h) d.) f(3h) Ex 2. Find 𝑔 𝑥+ℎ −𝑔(𝑥) ℎ a.) g(x) = 4𝑥 2 − 5𝑥 + 7 b.) g(x) = 𝑥 + 9 where h≠0 Assignment For the function f defined by evaluate each expression Graph of a function The graph of the function consists of all points (x, f(x)) where x is in the domain of f, note that x the directed distance from the y -axis and f(x) the directed distance from the x -axis. Equation: y = x² f(x) = 𝑥 − 2 g(x) = 𝑥² − 9 x2 + y2 = 25 The graphs of eight basic functions Finding the Domain and Range of a Function The domain of the function is the set of all -values for which which is the interval To find the range, observe that is never negative. Ex 3. Determine the Domain of the function g defined by g(x) = 𝑥(𝑥 − 2) Using Calculator MODE – 7 – TABLE Start: -4 End: 7 Step: 1 A Function Defined by More than One Equation For the piecewise-defined function Ex 4. Let f be the function defined by x-1 if x<3 f(x) = 5 if x=3 2x+1 if 3<x determine the domain and range Ex 5. Let g be the function g(x)= 3x-2 if x<1 x² if 1⦤x determine the domain and range Ex 6. The function h is defined by h(x) = 𝑥²−9 𝑥−3 determine the domain and range Ex 7. Let H be the function defined by H(x) = x + 3 if x ≠ 3 2 if x = 3 determine the domain and range Ex 8. The function of is defined by f(x) = x² if x ≠ 2 7 if x = 2 determine the domain and range Ex 9. Determine the domain and range of the function f for which f(x)= x f(x) = x if x ≥ 0 -x if x < 0 Assignment Ex 10. Determine the domain and range of the function f for which f(x) = x(1 – x2) Trigonometric Functions In mathematics, the trigonometric functions (also called circular functions, angle functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. Ex 11. Given ∅(x) = sin x Find: a. ∅(π) b. ∅(5/4 π) c. ∅(2/3 π) d. ∅(- π/6) e. ∅(π-y) Ex 12. H(Y) = cos 2y – sin y Find: H(0) H(-1/4 π) H(π) Ex 13. b−b² If F(b) = 1+b² Find : F(1/2) F(tan x) Assignment Ex. 14 If f(w) = 1 𝑤 − Find: f(cos x) 𝑤 1+sin 𝑥 Operation on Functions Given the two functions f and g: (i)Sum, denoted by : f+g defined by: (f+g)(x)=f(x)+g(x) (ii)Difference, denoted by f-g defined by: (f-g)(x)=f(x)-g(x) (iii)Product, denoted by f.g defined by: (f.g)(x)= f(x).g(x) (iv)Quotient, denoted by f/g defined by: (f/g) (x)= (f(x))/(g(x)) Example 1 f(x) = 2x - 3 g(x) = 𝑥 2 +1 find: a.) f + g b.) f g c.) g/f (v)Composite, denoted by f °g Defined by: (f °g)(x)=f(g(x)) Example 2 f(x)= 5 𝑥−2 g(x)= 2x+1 Find: (f °g)(3) in two methods Example 3 f(x)= 2x - 3 g(x) = cos x Find: f °f g°g f °g g °f Assignment Example 4 f(x)= sin x g(x) = π x Find: (f °g) (1/2) (g °f) (π /4) - Seatwork next meeting