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Functions, EXT1′ F1 2017 HSC 12a

Consider the function  `f(x) = (e^x - 1)/(e^x + 1)`.

  1.  Show that  `f(x)`  is increasing for all `x`.  (1 mark)

  2.  Show that  `f(x)`  is an odd function.  (1 mark)

  3.  Describe the behaviour of  `f(x)`  for large positive values of `x`.  (1 mark)

  4.  Hence sketch the graph of  `f(x) = (e^x - 1)/(e^x + 1)`.  (1 mark)

  5.  Hence, or otherwise, sketch the graph of  `y = 1/(f(x))`.  (1 mark)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
  3. `text{(See worked Solutions)}`
  4.  
  5.  

Show Worked Solution

i.  `text(Solution 1)`

`f(x) = (e^x – 1)/(e^x + 1)`

`u = e^x – 1, quadquad u prime = e^x`

`v = e^x + 1, quadquad v prime = e^x`
 

`f′(x)` `= {e^x (e^x + 1) – e^x (e^x – 1)}/(e^x + 1)^2`
  `= (e^(2x) + e^x – e^(2x) + e^x)/(e^x + 1)^2`
  `= (2e^x)/(e^x + 1)^2`

 
`text(S) text(ince)\ \ e^x > 0\ \ text(for all)\ \ x,`

`=>f prime (x) > 0\ text(for all)\ x`

`:. f(x)\ text(is increasing for all)\ x.`
 

`text(Solution 2)`

`f(x)` `= (e^x – 1)/(e^x + 1)`
  `= (e^x + 1 – 2)/(e^x + 1)`
  `= 1 – 2/(e^x + 1)`
`f′(x)` `= (2e^x)/(e^x + 1)^2`

 
`text(See Solution 1 for remainder.)`
 

ii.   `f(-x)` `= (e^(-x) – 1)/(e^(-x) + 1) xx e^x/e^x`
    `= (1 – e^x)/(1 + e^x)` 
    `= – (e^x – 1)/(e^x + 1`
    `= -f(x)`

 
`:. f(x)\ text(is odd.)`

 

iii.  `lim_(x -> oo) (e^x – 1)/(e^x + 1) = 1`

`text(i.e.  As)\ \ x -> oo,\ f(x) -> 1^-\ \ text{(i.e. from lower side)}`

 

iv.  

 

v.