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Q 5. Write the following partial function f: Z4 → Z4 in table form. f =...
Q 7. For each of the following functions Z4 + Z4, write down the inverse relation and decide whether or not it is a function Z4 Z4. Hence, decide which of these three functions, if any, are bijections. (a) fi = {(0,1),(1,2), (2,3), (3, 3)} RIS. (b) f2 = {(0, 1), (1, 2), (2,3), (3,0)} (c) f3 = {(0, 1), (1,0), (2, 3), (3, 2)}
= Q 6. Let f: {0,1} + {0,1} be a function given by f(0) = 0 = f(1). Find two (total) functions g: {0,1} +{0,1} and h: {0,1} +{0,1} such that fog #gof and foh= hof. Write out f and your chosen functions g and h in table form, using a single table. Only the table is required as the answer.
1. [2] Is the function f :Q\ {0} →Q defined by f(x) = 1 + 2 onto? Why or why not? 2. [3] Let A = {1, 2, 3, 4, 5,6}, and f: A+ A be the function given in the table below. 2 1 2 3 4 5 6 f(x) 3 5 6 24 1| (a) Explain why f is invertible. (b) Is it true that f-1 = f o f? Why or why not?
5. (a) Suppose f is any function with continuous second-order partial derivatives such that f(0,0) = 0 and (0,0) is a critical point of f. Write an expression for the second-degree Taylor polynomial, Q, of f at (0,0).
Find the directional derivative of the function f(x,y,z) = z4−x3y2 at P(1,-1,1) in the direction of the vector from P(1,-1,1) to the point Q(2,1,0). What is the maximum rate of change of f at the point P(1,-1,1) and in which direction
Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. *4 - 2x3 + x2 + 7x - 2 (a) x2 - 2x + 1
Write out the form of the partial fraction decomposition of the function (See Example). Do not determine the numerical values of the coefficients. A.) (x4 + 6) / x5 + 5x3 B.) 3 / (x2? 4)2
Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients.
8. Write out the form of the partial fraction expansion for the following transfer function. SOME FACTORING AND CANCELING MAY BE REQUIRED IF THE DENOMINATOR IS NOT IRREDUCIBLE G(s) = +4+46 +4+6) To get full credit you need to have the denominators correct and the form of the numer- ators correct. DO NOT solve for the values of the numerator coefficients. You don't need to for credit and it would take a long time. 8+2
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form b) Write the vertex form (also known as standard form). c) Find the y-intercept. Which of the three forms listed above is the most useful in identifying the y-intercept and why? d) Find the x-intercepts. Which of the three forms listed above is the most useful in identifying the x-intercepts and why? e) Find the vertex. Which of the three forms listed above is...