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Functions, 2ADV F2 2021 HSC 5 MC

Which of the following best represents the graph of  `y = 10 (0.8)^x`?
 

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`A`

Show Worked Solution

`\text{By elimination:}`

`\text{When} \ x = 0 \ , \ y = 10(0.8) ^0 = 10`

`-> \ text{Eliminate B and D}`

`text(As)\ \ x→oo, \ y→0`

`-> \ text{Eliminate C}`

`=> A`

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`