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Functions, 2ADV F1 SM-Bank 14 MC

Let  `g(x) = log_2(x),\ \ x > 0`

Which one of the following equations is true for all positive real values of  `x?`

A.   `2g (8x) = g (x^2) + 8`

B.   `2g (8x) = g (x^2) + 6`

C.   `2g (8x) = (g (x) + 8)^2`

D.   `2g (8x) = g (2x) + 6`

Show Answers Only

`B`

Show Worked Solution

`text(Consider Option)\ B:`

♦♦ Mean mark 35%.
`text(LHS)` `= 2g(8x)`
  `= 2log_2(8x)`
  `= 2log_2(8) + 2log_2(x)`
  `=2log_2 (2^3)+ 2log_2(x)`
  `= 6 + log_2(x^2)`
  `= g(x^2) + 6`

`=>   B`

Functions, 2ADV F1 SM-Bank 9 MC

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

A.   `3(x + 1)`

B.   `e^(3x) + 3`

C.   `e^(8x + 9)`

D.   `log_e (3x) + 3`

Show Answers Only

`A`

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

 
`=> A`

Functions, 2ADV F1 SM-Bank 7

Let  `f(x) = log_e(x)`  for  `x>0,`  and  `g (x) = x^2 + 1`  for  all `x`.

  1. Find  `h(x)`, where  `h(x) = f (g(x))`.  (1 mark)

  2. State the domain and range of  `h(x)`.  (2 marks)

  3. Show that  `h(x) + h(−x) = f ((g(x))^2 )`.  (2 marks)
Show Answers Only
  1. `log_e(x^2 + 1)`
  2. `text(Domain)\ (h):\ text(all)\ x`

     

    `text(Range)\ h(x):\ \  h>=0`

  3. `text(See Worked Solutions)`
Show Worked Solution
i.    `h(x)` `= f(x^2 + 1)`
    `= log_e(x^2 + 1)`

 

ii.   `text(Domain)\ (h) =\ text(Domain)\ (g):\ text(all)\ x`

♦♦ Mean mark part (a)(ii) 30%.
`=> x^2 + 1 >= 1`
`=> log_e(x^2 + 1) >= 0`

 
`:.\ text(Range)\ h(x):\ \  h>=0`

 

MARKER’S COMMENT: Many students were unsure of how to present their working in this question. Note the layout in the solution.
iii.   `text(LHS)` `= h(x) + h(−x)`
    `= log_e(x^2 _ 1) + log_e((−x)^2 + 1)`
    `= log_e(x^2 + 1) + log_e(x^2 + 1)`
    `= 2log_e(x^2 + 1)`

 

`text(RHS)` `= f((x^2 + 1)^2)`
  `= 2log_e(x^2 + 1)`

 
`:. h(x) + h(−x) = f((g(x))^2)\ \ text(… as required)`

Functions, 2ADV F1 SM-Bank 6 MC

Let  `f(x) = e^x + e^(–x).`

`f(2u)`  is equal to
  

A.   `f(u) + f(-u)`

B.   `2 f(u)`

C.   `(f(u))^2 - 2`

D.   `(f(u))^2 + 2`

Show Answers Only

`C`

Show Worked Solution

`text(By trial and error,)`

♦ Mean mark 44%.

`text(Consider)\ \ (f(u))^2 – 2:`

`f(2u)` `=e^(2u) + e^(-2u)`
`(f(u))^2` `=(e^u + e^(-u))^2`
  `=e^(2u) + 2 + e^(-2u)`
   

`:. f(2u) = (f(u))^2-2`

`=>C`

Functions, 2ADV F1 SM-Bank 5 MC

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

Then  `f(g(x))`  is given by
 

A.   `e^(4x + 6) + 2 e^(2x + 3) - 3`

B.   `2x^2 + 4x - 6`

C.   `e^(2x^2 + 4x - 3)`

D.   `e^(2x^2 + 4x - 6)`

Show Answers Only

`C`

Show Worked Solution

`text(By trial and error,)`

`text(Consider:)\ \ f(x) = e^(2x^2 + 4x – 3)`

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

 
`=> C`