MATHS 102 - Functioning in Mathematics
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MATHS 102 - Functioning in Mathematics
2022 Semester TWO
Assignment 1
1. (a) Given the expression a + b ÷ c x b - a x c for a, b, c e z and c 0, simplify to a single rational expression. Where there are common factors use the distributive property
as part of your simplification process.
(b) Determine whether or not the following statement is correct. If it is not, give a counter-example - that is, one example that shows the statement is NOT true. If you think it is true then write a proof.
The product of any two odd numbers is always odd.
(c) The expression is equivalent to . Find the fraction .
2. (a) If ^35a3 b simplifies to 7a^5a, what is the value of b?
(b) Use the rule: (a3 - b ) = (a3 - b)(a2 + ab + b ) to fully factorise 512x218 - y27 .
(c) Tama and Suzie are working on simplifying the expression + to a single fraction with a rational denominator. Tama says, the two fractions must be rationalised first. Suzie says, the two fractions must be combined before rationalising the result.
Show that both Tama and Suzie’s strategies give the same simplified form of the expression.
3. A quadratic equation is given by 3x2 + kx - 2 = 0.
(a) What value(s) of k would give the equation 2 real solutions?
(b) If particular solutions to the equation were x = -1 and x = , what would be the
value of k .
4. (a) Solve the following inequalities and represent the solutions in either interval or set notation.
(i) x - 6 < s x +
1 1
(ii) + < 2
(b) In a small pastry shop, the cost of producing x number of pastry items for catering orders is given by the equation C(x) = 3x +35, where C is the cost in dollars. The shop wants to give 2 items of their new pastry products as a complimentary for large orders but only if the average cost per item (including the 2 free items) is less than $5.
(i) Write an expression for the inequality representing the average cost of less than $5. Solve the inequality to find the number of pastry items that should be in an order to guarantee receiving the shop’s 2 complimentary items. Interpret your answer.
Hint: The average cost of making x items plus the extra 2 free pastry items is given by
3x + 35
(ii) Explain why considering the undefined value of x to be a critical value in this particular scenario not necessary.
5. Consider the system of two lines below:
3x + ky = 1
x - 4y = l
By solving the system simultaneously, what value(s) of k and l could make the two lines:
(a) meet at a single point.
(b) have no intersection.
(c) coincide.
6. Consider the three functions f (x) = x2 , g(x) = ^x and h(x) = and answer the following
questions:
(a) Is it possible to construct a composite function, using all the three functions and using a function only once, to create a composite function that has a domain R? If yes, give the
composite function. If no, give an explanation.
(b) Give the domain and range of:
(i) h(h(x))
(ii) f (g(f (x)))
2022-08-05