Mathematics Dictionary Dr. K. G. Shih

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Functions

Subjects

Read Symbol defintion Q01 | - Defintion Q02 | - Function names Q03 | - Properties of functions Q04 | - Composite functions Q05 | - Factors of functions Q06 | - Inverse functions Q07 | - Rationalize denominator Q08 | - Reference Q09 | - Q10 | -

Answers

Q01. Defintion An expression has one variable has replation with other variable. Expression : y = F(x). Independent variable is x. It is also called domain. Dependent variable is y. It is also called range. For y = F(x), it has only one y value for each given x value. y = Sqr(1 - x^2) is a function. x^2 + y^2 = 1 is not a functione because for each x value y has 2 values. Go to Begin Q02. Name of functions By degree Constant function : y = c where c is constant. It is a horizontal line Linear function : y = a*x + b. It is a line. Quadratic functions : y = a*x^2 + b*x + c. It is a parabola Cubic function : y = a*x^3 + b*x^2 + c*x + d. Quartic function : y = a*x^4 + b*x^3 + c*x^2 + d*x + e. By expression form Explicit function : Function y = F(x) is explicit function. Example : y = x^2 - 6*x + 8 is explicit function. Implicit function : F(x,y) = 0 is implicit funtion. Example : x^2 - 6*x + 8 - y = 0 is implicit funtion. Rational function y = F(x)/G(x). Example : (x+2)/(x^2-6*x+8) Irrational function y = Sqr(F(x)). Example : y = Sqr(1 - x^2) Complex function z = x + i*y. Go to Begin Q03. Properties of cuntions Odd functions If F(-x) = -F(x), F(x) is odd function. Example : F(x) = x^3 is odd function since F(-x) = (-x)^3 = -x^3 = -F(x). Property : Rotate 180 degree about origin, the curve will be same. Even functions If F(-x) = F(x), F(x) is evend function. Example : F(x) = x^2 is even function since F(-x) = (-x)^2 = x^2 = F(x). Property : The curve is symmetrical to y-axis. Periodic functions If F(x+p) = F(x), F(x) is periodic function with period p. Example 1 : sin(x + 2*pi) = sin(x) and period is 2*pi. Example 2 : cos(x + 2*pi) = cos(x) and period is 2*pi. Example 3 : tan(x + 1*pi) = tan(x) and period is 1*pi. Go to Begin Q04. Composite function Defintion y = F(G(x)) is composite function where G(x) is also a fuction. Example : F(x) = x^2 + 2*x - 4 and G(x) = x + 1, find F(G(a)) F(G(a)) = (G(a))^2 + 2*G(a)x - 4 F(G(a)) = (a + 1)^2 + 2*(a+1) - 4 Composite of inverse e^(ln(x)) = x. ln(e^x) = x. sin(arcsin(x)) = x. arcsin(sin(A)) = A. Go to Begin Q05. Factors of Expressions x^2 - y^2 = (x+y)*(x-y) This is square differen equals sum times difference. x^2 + y^2 = no real factors. x^3 - y^3 = (x-y)*(x^2 + x*y +y^2). x^3 + y^3 = (x+y)*(x^2 - x*y +y^2). x^4 - y^4 = (x+y)*(x-y)*(x^2+y^2). x^4 + y^4 = no real factors. x^5 - y^5 = (x-y)*(x^4 + (x^3)*y +(x^2)*(y^2) - x*(y^3) + y^3. x^5 + y^5 = (x+y)*(x^4 - (x^3)*y +(x^2)*(y^2) + x*(y^3) + y^3.. Factors in perfect square x^2 - 2*x*y + y^2 = (x-y)^2 x^2 + 2*x*y + y^2 = (x+y)^2 Factors in perfect cube x^3 - 3*(x^2)*y + 3*x*(y^2) - y^3 = (x-y)^3 x^3 + 3*(x^2)*y + 3*x*(y^2) + y^3 = (x+y)^3 Studys Subject | Factors of expression of linear functions Go to Begin Q06. Inverse functions Studys Subject | Inverse of linear functions Studys Subject | Inverse of Quadratic functions Go to Begin Q07. Rationalize denominator Example 1 : Rationalize y = 1/Sqr(1-x^2) Make denominator a rational function. Hence multiply numerator and denominator by Sqr(1-x^2). y = Sqr(1-x^2)/(Sqr(1-x^2)*Sqr(1-x^2). y = Sqr(1-x^2)/(1-x^2). Example 1 : Rationalize y = 1/(2+i) Make denominator a rational function. Hence multiply numerator and denominator by (2-i). y = (2+i)/((2+i)*(2-i)). y = (2+i)/(2^2 - i^2). Since i^2 = -1. Hence y = (2+i)/5. Go to Begin Q08. Reference Studys Subject | Sketch y = F(x) or y = F(x)/G(x) Click start. Click Subject in upper box Subject 1 : Linear functions. Subject 2 : Quadratic functions. Subject 3 : Rational functions. Click program in lower box Studys Subject | Complex numbers Go to Begin Q09. Answer Go to Begin Q10. Answer Go to Begin

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