smc-6216-10-Log/Exp

Functions, 2ADV F1 EQ-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?`

`2g (8x) = g (x^2) + 8`
`2g (8x) = g (x^2) + 6`
`2g (8x) = (g (x) + 8)^2`
`2g (8x) = g (2x) + 6`

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`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 EQ-Bank 3 MC

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

`3(x + 1)`
`e^(3x) + 3`
`e^(8x + 9)`
`log_e (3x) + 3`

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`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 EQ-Bank 26

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

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

--- 2 WORK AREA LINES (style=lined) ---
State the domain and range of `h(x)`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Show that `h(x) + h(−x) = f ((g(x))^2 )`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---

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a. `log_e(x^2 + 1)`

b. text(Domain)\ (h):\ text(all)\ x`

`text(Range)\ h(x):\ \ h>=0`

c. `text(See Worked Solutions)`

Show Worked Solution

a.	`h(x)`	`= f(x^2 + 1)`
		`= log_e(x^2 + 1)`

b. `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.

c.	`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 EQ-Bank 16 MC

Let `f(x) = e^x + e^(-x).`

`f(2u)` is equal to

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

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`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 EQ-Bank 1 MC

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

Then `f(g(x))` is given by

`e^(4x + 6) + 2 e^(2x + 3)-3`
`2x^2 + 4x-6`
`e^(2x^2 + 4x-3)`
`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`