Algebra and function
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This document contains sample algebra and function questions with answers: 1) It finds the sum of f+g and fg, where f(x)=1/x^2 and g(x)=sqrt(x). 2) It solves an inequality involving rational functions and determines the solution range. 3) It finds the inverse of a 3x3 matrix using the ERO method. 4) It provides answers to additional questions about domains, ranges, function composition, sequences, systems of equations, and classifying number sets.
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![ALGEBRA AND FUNCTION
1
Part B: Q1: Let f(x)=1/x^2, and g(x)=sqrt(x). Determine:
Q1a)f+g and fg
Answers: f+g =1/x^2+ sqrt(x)=(1+sqrt(x^5))/x^2. fg= [1/x^2]
sqrt(x)=1/sqrt(x^3).
Q5: Obtain the sum of the first 10 terms: {2,5,8,11,14,……}.
Answer: Using formulae Sn = n/2[ 2a+(n-1)d], when a=2, d=3 and n=10, S10 =
10/2[ 2(2)+(10-1)3]=155.](https://image.slidesharecdn.com/algebraandfunction-180719113040/85/Algebra-and-function-1-320.jpg)
![ALGEBRA AND FUNCTION
2
Q2, Solve 1/(x+1) > 1/(x-1).
Answer: 1/(x+1)- 1/(x-1) > 0, (x-1)-(x+1)/(x+1)(x-1) >0, -2/(x+1)(x-1)>0,For x< -
1, we gain negative value, for -1<x<1, we gain positive value and for x>1, we gain
negative value. So, for 1/(x+1)- 1/(x-1) > 0, we choose positive value, then the
solution is in the range -1< x < 1.
Q3, determine the inverse matrix of [0 0 1, 1 0 0, 0 1 0].
Answer: By using ERO method, [ A / I ] =[0 0 1, 1 0 0, 0 1 0 / 1 0 0, 0 1 0, 0 0 1], A12….
[1 0 0, 0 0 1, 0 1 0 / 0 1 0, 1 0 0, 0 0 1],
A23…[1 0 0, 0 1 0, 0 0 1 / 0 1 0, 0 0 1, 1 0 0]= [ I / A-1],
So A-1 = [0 1 0, 0 0 1, 1 0 0].](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
1. functions
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
GEN-MATH-WEEK-2.pptx
This document provides an agenda and examples for performing operations on functions such as addition, subtraction, multiplication, division, and composition of functions. It gives the definitions and formulas for summing, differencing, multiplying, and dividing two functions f and g. Examples are worked out applying these operations to specific functions f(x) = 4x + 3 and g(x) = 3x - 2. The document also defines composition of functions using circle notation and provides examples of evaluating compositions such as f(g(x)) and g(f(x)).
Newton two Equation method
This document describes an experiment using the Newton-Raphson method to find the roots of nonlinear equations in MATLAB. Two nonlinear equations are given as an example: x^2+xy=10 and y+3xy^2=57. The MATLAB code implements the Newton-Raphson method to iteratively calculate the roots. For the given equations, the method converges after 15 iterations with roots of x=4.3937 and y=-2.1178. The experiment demonstrated the use of the Newton-Raphson method to solve nonlinear equations numerically in MATLAB.
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Algebra and function
- 1. ALGEBRA AND FUNCTION 1 Part B: Q1: Let f(x)=1/x^2, and g(x)=sqrt(x). Determine: Q1a)f+g and fg Answers: f+g =1/x^2+ sqrt(x)=(1+sqrt(x^5))/x^2. fg= [1/x^2] sqrt(x)=1/sqrt(x^3). Q5: Obtain the sum of the first 10 terms: {2,5,8,11,14,……}. Answer: Using formulae Sn = n/2[ 2a+(n-1)d], when a=2, d=3 and n=10, S10 = 10/2[ 2(2)+(10-1)3]=155.
- 2. ALGEBRA AND FUNCTION 2 Q2, Solve 1/(x+1) > 1/(x-1). Answer: 1/(x+1)- 1/(x-1) > 0, (x-1)-(x+1)/(x+1)(x-1) >0, -2/(x+1)(x-1)>0,For x< - 1, we gain negative value, for -1<x<1, we gain positive value and for x>1, we gain negative value. So, for 1/(x+1)- 1/(x-1) > 0, we choose positive value, then the solution is in the range -1< x < 1. Q3, determine the inverse matrix of [0 0 1, 1 0 0, 0 1 0]. Answer: By using ERO method, [ A / I ] =[0 0 1, 1 0 0, 0 1 0 / 1 0 0, 0 1 0, 0 0 1], A12…. [1 0 0, 0 0 1, 0 1 0 / 0 1 0, 1 0 0, 0 0 1], A23…[1 0 0, 0 1 0, 0 0 1 / 0 1 0, 0 0 1, 1 0 0]= [ I / A-1], So A-1 = [0 1 0, 0 0 1, 1 0 0].
- 3. ALGEBRA AND FUNCTION 3 Q1c) the domain and the range of g-1. Answers: For g-1, domain is x is all real numbers and range is g-1(x)>or =0. Q1b) f o g Answers: f o g = 1/(sqrt(x))^2=1/x
- 4. ALGEBRA AND FUNCTION 4 No 5: Given the sequence n^2. (a) write the 5 th term. Answer: T 5 = 5^2= 25. No 4: Write the system in the form of matrix equation. x+ z=0, y+ z=0, x+y=0. Answer: [ 1 0 1, 0 1 1, 1 1 0] [x , y, z] = [ 0, 0, 0 ].
- 5. ALGEBRA AND FUNCTION 5 (b)Determine its inverse. its inverse is the matrix itself , ie [ 1 0, 0 1]. Note: I x I = I. (a)Write the identity matrix. Answer: the identity matrix 2 x 2 is [ 1 0, 0 1],
- 6. ALGEBRA AND FUNCTION 6 Solve (x+1)/(x+2)(x+1) >0. Answer: The roots are -1 or -2. Consider the interval, -2<x, -2<x<-1, x>-1. Construct the interval table: -2<x -2<x<-1 x>-1. (x+1) - - + (x+2) - + + (x+1) - - + Result - + + Since LHS >0, choose the positive value. Conclusion : the solution is x> -2.
- 7. ALGEBRA AND FUNCTION 7 (d) All integers are natural numbers. Answer: FALSE. ( Not all integers are natural numbers, example: integer -2 is not a natural number ) ( c) All natural numbers are integers. Answer: TRUE. ( Set of natural numbers is a subset of set of intergers).
- 8. ALGEBRA AND FUNCTION 8 (b) All rational numbers are integers. Answer: FALSE. ( Not all rational numbers are intergers , example: rational number 1.5=3/2 is a fraction ) Part A No1: (a) All intergers are rational numbers. Answer: TRUE. ( Set of intergers is a subset of set of rational numbers)