HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

CONTENT POWER FUNCTION 2- ROOT FUNCTION 3- RATIONAL FUNCTION 4- TRIGONOMETRIC FUNCTION 5- EXPONENTIAL FUNCTION 6- LOGARITHMIC FUNCTION 7- INVERSE FUNCTION: TRIGONOMETRIC 8- HYPERBOLIC FUNCTION

Power Function The function y=x a, where a is a constant is called a power function (i) When a=n, a positive integer, the graph of f is similar to the parabola y=x 2 if n is even and similar to the graph of y=x 3 if n is odd However as n increases, the graph becomes flatter near 0 and steeper when  x   1

The graphs of x 2, x 4, x 6 on the left and those of x 3, x 5 on the right

(ii) a=1/n, where n is a positive integer Then is called a root function Root functions if n is even if n is odd The graph of f is similar to that of if n is even and similar to that of if n is odd

(1,1)

(iii) When a=–1, is the reciprocal function The graph is a hyperbola with the coordinate axes as its asymptotes

Rational functions A rational function is the ratio of two polynomials: is a rational function whose domain is {x/x  0} Where P and Q are polynomials. The domain of f consists of all real number x such that Q(x)  0.

Domain(f)={x/ x   2}

Trigonometric functions sinx and cosx are periodic functions with period 2  : sin(x + 2  ) = sinx, cos(x + 2  ) = cosx, for every x in R the domains of sinx and cosx are R, and their ranges are [-1,1] f(x)=sinx g(x)=cosx

These are functions of the form f(x)=a x, a > 0 Exponential functions y=2 x y=(0.5) x

Logarithmic functions These are functions f(x)=log a x, a > 0. They are inverse of exponential functions log 2 x log 3 x log 10 x log 5 x

Definition. A function f is a one-to-one function if: x 1  x 2  f(x 1 )  f(x 2 ) f g f is one-to-one g is not one-to-one : 2  3 but g(2) = g(3)

Example. Is the function f(x) = x 3 one-to-one ? Solution1. If x 1 3 = x 2 3 then (x 1 – x 2 )(x x 1 x 2 + x 2 2 ) = 0  x 1 = x 2 because hence f(x) = x 3 is one-to-one

Definition. Let f be a one-to-one function with domain A and range B. Then the inverse function f - 1 has domain B and range A and is defined by: domain( f –1 ) = range (f) range(f -1 ) = domain(f) f -1 (y) = x  f(x) = y, for all y in B Inverse functions

f Example. Let f be the following function AB

f -1 Then f -1 just reverses the effect of f AB

f -1 (f(x)) = x, for all x in A f(f -1 (x)) = x, for all x in B If we reverse to the independent variable x then: f -1 (x) = y  f(y) = x, for all x in B How to find f –1 Step1 Write y = f(x) Step2 Solve this equation for x in terms of y Step3 Interchange x and y. The resulting equation is y = f -1 (x)

Example. Find the inverse function of f(x) = x Solution. First write y = x Then solve this equation for x: Interchange x and y:

Question: When the trigonometric funtion y = sinx is one – to – one and how about its inverse function? Inverse trigonometric functions Application: Compute the integral

Considering analogicaly for the functions y = cosx, y = tgx, y = cotgx, we give the definition of three others inverse trigonometric functions Inverse trigonometric functions Application: Compute the integral

The four next functions are called hyperbolic function Hyperbolic functions We get directly hyperbolic formulas from all familiar trigonometric formulas by changing cosx to coshx and sinx to isinhx (i: imaginary number, i 2 = –1)

Hyperbolic formulas Application: Compute the integral

Piecewise defined functions 1 1 f(0)=1-0=1, f(1)=1-1=0 and f(2)=2 2 =4 The graph consists of half a line with slope –1 and y-intercept 1; and part of the parabola y = x 2 starting at the points (1,1) (excluded)