# Functions

Calculators are allowed for all questions unless indicated by the question. There are 5 questions for this topic. Alternatively, you can download the question paper and mark scheme using the links below.

Question 1:

Simplify the following:

A. $4c&space;\times&space;4c^3$ (1 mark)

B. $x^^{\frac{1}{2}}&space;\times&space;4x^{-\frac{1}{2}}$ (2 marks)

C. $ax^2&space;\times&space;a^3x$ (2 marks)

$16c^{4}$ (1 mark)

$x^{\frac{1}{2}}&space;\times&space;x^{-\frac{1}{2}}&space;=&space;1$ (1 mark)

$4(x^{\frac{1}{2}}&space;\times&space;x^{-\frac{1}{2}}&space;)=&space;4$ (1 mark)

$a&space;\times&space;a^{3}&space;=&space;a^{4}$ , $x^{2}&space;\times&space;x&space;=&space;x^{3}$ (1 mark)

$=a^{4}x^{3}$ (1 mark)

Question 2:

$g(x)&space;=&space;5x^{2}&space;+&space;2$

Calculate the following values:

A. $g(2x)$ (2 marks)

B. $g(x+5)$ (3 marks)

C. Solve the equation $g(x+5)&space;=&space;22$  for x. Giving your answer in surd form. (2 marks)

$5(2x)^{2}&space;+&space;2$ (1 mark)

$5(2x)^{2}&space;=&space;5(4x^{2})&space;=&space;20x^{2}$ (1 mark)

$=&space;20x^{2}&space;+&space;2$ (1 mark)

$g(x+5)=4(x+5)^{2}&space;+&space;2$ (1 mark)

$4(x+5)^2&space;=&space;4(x^2&space;+&space;10x&space;+&space;25)&space;=&space;4x^2&space;+&space;40x&space;+&space;100$ (1 mark)

$4x^2&space;+&space;40x&space;+&space;100&space;+&space;2&space;=&space;4x^2&space;+40x&space;+102$ (1 mark)

$\\4x^2&space;+&space;40x&space;+&space;102&space;=&space;22\\&space;4x^2&space;+40x&space;+&space;80&space;=&space;0$

(1 mark)

$x^2&space;+&space;10x&space;+&space;20&space;=&space;0$

$x&space;=&space;\pm&space;\sqrt5&space;-5$ (1 mark)

Question 3:

$\\g(x)=&space;(x+6)(x-1)\\&space;f(x)&space;=&space;2x-5$

A. Calculate the value of $g(f(x))$ (2 marks)

B. Solve the equation $g(f(x))$ = 0 (2 marks)

$g(f(x))&space;=&space;((2x-5)&space;+&space;6)((2x-5)&space;-&space;1)&space;=&space;(2x+1)(2x-6)$ (1 mark)

$(2x+1)(2x-6)=&space;4x^2&space;-10x&space;-&space;6$ (1 mark)

$4x^2&space;-10x&space;-&space;6&space;=&space;0$ (1 mark)

$x&space;=&space;-\frac{1}{2},&space;x=&space;3$ (1 mark)

Question 4:

You are given a rectangle with length $x+3$ and $x+5$ respectively.

A. Calculate the area of the rectangle in terms of $x$ (2 marks)

B. The side with length $x+3$ increases by 30%. Find the new area of the rectangle in terms of $x$. (3 marks)

C. Using $x=4$, calculate the new perimeter of the rectangle. (1 mark)

The area is equal to $(x+3)(x+5)$ (1 mark)

$(x+3)(x+5)=&space;x^2&space;+8x&space;+&space;15$ (1 mark)

New side length is $1.3(x+3)$ (1 mark)

New area is $1.3(x+3)(x+5)=&space;1.3(x^2&space;+8x&space;+&space;15)=&space;1.3x^2&space;+&space;10.4x&space;+&space;19.5$ (2 marks)

$2(1.3\times(4+3))&space;+&space;2(4+5)&space;=&space;36.2$ (1 mark)

Question 5:

James reads 13 pages of his book every day, there are 20 words on each page. After 30 days, there are 25 words left.

A. How many words are in the book? (2 marks)

B. Write an expression for the number of words in a book when someone reads $x$ pages per day, there are $y$ words on every page, and after $z$ weeks, there are only $n$ words left. (2 marks)

$13&space;\times&space;20&space;\times&space;30&space;=&space;7800$ (1 mark)

Total words in the book:

$7800&space;+&space;25&space;=&space;7825$ (1 mark)

Words they have read after $z$ weeks:

$=&space;7xyz$ (1 mark) [there are $7xy$ words read in 1 week]

Total words in the book:
$=7xy&space;+&space;n$ (1 mark)

$2(1.3\times(4+3))&space;+&space;2(4+5)&space;=&space;36.2$ (1 mark)