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Functions, 2ADV F2 SM-Bank 35

  1. Sketch the function  `y = f(x)`  where  `f(x) = (x - 1)^3`  on a number plane, labelling all intercepts.  (1 mark)

  2. On the same graph, sketch  `y = −f(−x)`. Label all intercepts.  (2 marks)

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  1.   
  2.   
Show Worked Solution

i.   `y = (x – 1)^3 => y = x^3\ text(shifted 1 unit to the right.)`
 

 

ii.   `y = −f(x) \ => \ text(reflect)\ \ y = (x – 1)^3\ \ text(in)\ xtext(-axis).`

`y = −f(−x) \ => \ text(reflect)\ \ y = −f(x)\ \ text(in)\ ytext(-axis).`

 

Functions, 2ADV F2 SM-Bank 13

 

  1. Show that the function  `y = (1-e^x)/(1 + e^x)`  is an odd function?   (1 mark) 

  2. Sketch  `y = (1-e^x)/(1 + e^x)`, labelling all intercepts and asymptotes.   (2 marks)

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  1. `{text(all real)\ x,  x!=1/4}`

 

 

 

 

 

 

Show Worked Solution

i.   `f(x) = (1-e^x)/(1 + e^x)`

`f(−x)` `= (1-e^(−x))/(1 + e^(−x)) xx (e^x)/(e^x)`
  `= (e^x-1)/(e^x + 1)`
  `= −(1-e^x)/(1 + e^x)`
  `= −f(x)`

 
`:. f(x)\ text(is ODD.)`

 

ii.   `y = (1-e^x)/(1 + e^x) xx (e^(−x))/(e^(−x)) = (e^(−x)-1)/(e^(−x) + 1) = 1-2/(e^(−x) + 1)`

`text(As)\ x -> ∞, \ 2/(e^(−x) + 1) -> 2, \ y -> −1`

`text(As)\ x ->-∞, \ 2/(e^(−x) + 1) -> 0, \ y -> 1`

`text(When)\ x = 0, \ y = 0`
 

Functions, 2ADV F2 SM-Bank 10

Consider the function  `f(x) = x/(4 - x^2)`.

  1. Identify the domain of  `f(x)`.   (1 mark)

  2. Sketch the graph  `y = f(x)`, showing all intercepts and asymptotes.   (3 marks)

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  1. `text(Domain:)\ \ text(all)\ x,\ x != +- 2`
  2.  
     
Show Worked Solution

i.   `4-x^2 !=0`

`=> x!= 2 or -2`

`:.\ text(Domain:)\ \ text(all)\ x,\ x != +- 2`

 

ii. `text(Asymptotes at)\ \ x = +- 2`

`text(Passes through)\ (0,0)`

COMMENT: Note that  `x->2^(–)`  is a short way of writing as `x` approaches 2 from the negative (or left-hand) side. This notation can save time when required.
`text(As)` `\ \ x -> 2^(-),` `\ y -> oo`
  `\ \ x -> 2 ^(+),` `\ y -> – oo`
`text(As)` `\ \ x -> -2^ (-),` `\ y -> oo`
  `\ \ x -> -2^ (+),` `\ y -> -oo`
`text(As)` `\ \ x -> oo,` `\ y -> 0`
  `\ \ x -> – oo,` `\ y -> 0`

 

  EXT1 2013 11d