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Algebra, MET2 2007 VCAA 2 MC

Let  `g(x) = x^2 + 2x - 3 and f(x) = e^(2x + 3).`

Then  `f(g(x))`  is given by 

  1. `e^(4x + 6) + 2 e^(2x + 3) - 3`
  2. `2x^2 + 4x - 6`
  3. `e^(2x^2 + 4x + 9)`
  4. `e^(2x^2 + 4x - 3)`
  5. `e^(2x^2 + 4x - 6)`
Show Answers Only

`D`

Show Worked Solution

`text(Solution 1)`

`text(Define)\ \ f(x) and g(x)\ \ text(on CAS)`

`f(g(x)) = e^(2x^2 + 4x – 3)`

`=>   D`
 

`text(Solution 2)`

`f(g(x))` `=e^(2 xx (x^2 + 2x – 3)+3)`
  `= e^(2x^2 + 4x – 3)`

`=>D`

Algebra, MET2 2010 VCAA 4 MC

If  `f(x) = 1/2e^(3x)  and  g(x) = log_e(2x) + 3`  then  `g (f(x))` is equal to
 

  1. `2x^3 + 3`
  2. `e^(3x) + 3`
  3. `e^(8x + 9)`
  4. `3(x + 1)`
  5. `log_e (3x) + 3`
Show Answers Only

`D`

Show Worked Solution

`text(Define)\ \ f(x)= 1/2e^(3x), \ g(x)= log_e(2x) + 3`

`g(f(x))` `= log_e(2 xx 1/2e^(3x)) + 3`
  `=log_e e^(3x) + 3`
  `=3x + 3`
  `= 3 (x + 1)`

 
`=>   D`