smc-986-30-Other

Functions, 2ADV F1 EQ-Bank 18

If \(f(x)=x^2-3\) and \(g(x)=\sqrt{x-2}\),

Find \(f(g(x))\). (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
What is the domain of \(f(g(x))\) ? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. \(f(g(x))=x-5\)

b. \(\text{Restriction for}\ \ g(x)\ \Rightarrow \ x-2\geqslant 0 \ \Rightarrow \ x \geqslant 2\)

\(\text{Domain}\ \ f(g(x)):\ x \geqslant 2\)

Show Worked Solution

a.	\(f(g(x))\)	\(=(\sqrt{x-2})^2-3\)
		\(=x-2-3\)
		\(=x-5\)

b. \(\text{Restriction for}\ \ g(x)\ \Rightarrow \ x-2\geqslant 0 \ \Rightarrow \ x \geqslant 2\)

\(\text{Domain}\ \ f(g(x)):\ x \geqslant 2\)

Functions, 2ADV F1 2022 HSC 10 MC

The graphs of `y=f(x)` and `y=g(x)` are shown.

Which graph best represents `y=g(f(x))`

Show Answers Only

`B`

Show Worked Solution

`text{By Elimination:}`

`y=f(x)\ \ text{is an even function}`

`y=f(x)=f(-x)\ \ =>\ \ g(f(x))=g(f(-x))`

`g(f(x))\ \ text{is also an even function}`

`text{→ Eliminate A, C and D}`

`=>B`

♦ Mean mark 46%.

Functions, 2ADV F1 SM-Bank 16 MC

Let `f(x)` and `g(x)` be functions such that `f(–1)=4, \ f(2)=5, \ g(–1)=2, \ g(2)=7` and `g(4)=6`.

The value of `g(f(–1))` is

Show Answers Only

`D`

Show Worked Solution

`f(–1)=4`

`g(f(–1)) = g(4) = 6`

`=>D`

Functions, 2ADV F1 EQ-Bank 11

Given the function `f(x) = sqrt(3 - x)` and `g(x) = x^2 - 2`, sketch `y = g(f(x))` over its natural domain. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

`g(x) = x^2 – 2,\ \ f(x) = sqrt(3 – x)`

`g(f(x))`	`= (sqrt(3 – x))^2 – 2`
	`= 3 – x – 2`
	`= 1 – x`

`text(S)text(ince)\ \ f(x) = sqrt(3 – x),`

`=> text(Domain:)\ x <= 3`

Functions, 2ADV F1 SM-Bank 30

Given `f(x) = sqrtx` and `g(x) = 25 - x^2`

Find `g(f(x))`. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Find the domain and range of `f(g(x))`. (2 marks)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`25 – x`
`text(Domain:)\ −5<= x <= 5`

`text(Range:)\ 0<=y<= 5`

Show Worked Solution

i.	`g(f(x))`	`= 25 – (f(x))^2`
		`= 25 – (sqrtx)^2`
		`= 25 – x`

ii.	`f(g(x))`	`= sqrt(g(x))`
		`= sqrt(25 – x^2)`

`:.\ text(Domain:)\ −5<= x <= 5`

`:.\ text(Range:)\ 0<=y<= 5`

Functions, 2ADV F1 SM-Bank 15 MC

If the equation `f(2x) - 2f(x) = 0` is true for all real values of `x`, then `f(x)` could equal

A. `x^2/2`

B. `sqrt (2x)`

C. `2x`

D. `x - 2`

Show Answers Only

`C`

Show Worked Solution

`text(We need)\ \ f(2x)=2\ f(x),`

`text(Consider)\ C,`

`f(x)`	`=2x,`
`f(2x)`	`= 2(2x)`
	`= 2\ f(x)`

`=> C`

Functions, 2ADV F1 SM-Bank 13 MC

Which one of the following functions satisfies the functional equation `f (f(x)) = x`?

A. `f(x) = 2 - x`

B. `f(x) = x^2`

C. `f(x) = 2 sqrt x`

D. `f(x) = x - 2`

Show Answers Only

`A`

Show Worked Solution

`text(By trial and error,)`

`text(Consider:)\ \ f(x)=2-x`

`f(f(x))`	`=2-(2-x)`
	`=x`

`=> A`

Functions, 2ADV F1 SM-Bank 11

Given `f(x) = sqrt (x^2 - 9)` and `g(x) = x + 5`

Find integers `c` and `d` such that `f(g(x)) = sqrt {(x + c) (x + d)}` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
State the domain for which `f(g(x))` is defined. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`c = 2, d = 8 or c = 8, d = 2`
`x in (– oo, – 8] uu [– 2, oo)`

Show Worked Solution

a.	`f(g(x))`	`= sqrt {(x + 5)^2 – 9}`
		`= sqrt (x^2 + 10x + 16)`
		`= sqrt {(x + 2) (x + 8)}`

`:. c = 2, d = 8 or c = 8, d = 2`

b. `text(Find)\ x\ text(such that:)`

♦♦♦ Mean mark 13%.
MARKER’S COMMENT: “Very poorly answered” with a common response of `-3 <=x<=3` that ignored the information from part (a).

`(x+2)(x+8) >= 0`

`(x + 2) (x + 8) >= 0\ \ text(when)`

`x <= -8 or x >= -2`

`:.\ text(Domain:)\ \ x<=-8\ \ and\ \ x>=-2`

Functions, 2ADV F1 SM-Bank 4 MC

The function `f(x)` satisfies the functional equation `f (f (x)) = x` for `{x:\ text(all)\ x,\ x!=1}`.

The rule for the function is

A. `f(x) = x + 1`

B. `f(x) = x - 1`

C. `f(x) = (x - 1)/(x + 1)`

D. `f(x) = (x + 1)/(x - 1)`

Show Answers Only

`D`

Show Worked Solution

`text(By trial and error:)`

♦ Mean mark 47%.

`text(Consider)\ \ f(x) = (x + 1)/(x – 1)`

`f(f(x))`	`=((x + 1)/(x – 1) +1)/((x + 1)/(x – 1) -1)`
	`=(x+1+x-1)/(x+1-x+1)`
	`=x`

`=>D`

Functions, 2ADV F1 SM-Bank 3

Let `f(x) = sqrt(x + 1)` for `x>=0`

State the range of `f(x)`. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---
Let `g(x)=x^2+4x+3`, where `x<=c` and `c<=0`.
Find the largest possible value of `c` such that the range of `g(x)` is a subset of the domain of `f(x)`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`y>= 1`
`-3`

Show Worked Solution

i. `text(Sketch of)\ \ f(x):`

`:.\ text(Range:)\ \ y>= 1`

ii. `text(Sketch)\ \ g(x) = (x + 1) (x + 3)`

`text(Domain of)\ \ f(x):\ \ x>=0`

`text(Find domain of)\ g(x)\ text(such that range)\ g(x):\ y>=0`

`text(Graphically, this occurs when)\ \ g(x)\ \ text(has domain:)`

`x <= –3\ \ text(and)\ \ x>=-1`

`:. c = -3`

Functions, 2ADV F1 SM-Bank 2 MC

Let `f(x)` and `g(x)` be functions such that `f (2) = 5`, `f (3) = 4`, `g(2) = 5`, `g(3) = 2` and `g(4) = 1`.

The value of `f (g(3))` is

Show Answers Only

`D`

Show Worked Solution

`f(g(3))`	`=f(2)`
	`=5`

`=> D`

Functions, 2ADV F1 SM-Bank 1 MC

Let `h(x) = 1/(x - 1)` for `-1<h<1`.

Which one of the following statements about `h` is not true?

`h(x)h(–x) = –h(x^2)`
`h(x) - h(0) = xh(x)`
`h(x) - h(–x) = 2xh(x^2)`
`(h(x))^2 = h(x^2)`

Show Answers Only

`D`

Show Worked Solution

`text(By trial and error, consider option)\ E:`

♦ Mean mark 46%.

`h(x) = 1/(x – 1)`

`(h(x))^2 = 1/(x – 1)^2=1/(x^2-2x+1)`

`h(x^2)=1/(x^2-1) != (h(x))^2`

`=> D`