SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Functions, 2ADV F1 2025 HSC 18

Find the range of \(g(f(x))\),  given  \(f(x)=\dfrac{3}{x-1}\)  and  \(g(x)=x+5\).   (2 marks)

Show Answers Only

\(\text{Range} \ g(f(x)): \text{All} \  y, \  y \neq 5 \ \ \text{or} \ \ \{-\infty<y<5\} \cup\{5<y<\infty\}.\)

Show Worked Solution

\(f(x)=\dfrac{3}{x-1}, \ \ g(x)=x+5\)

\(g(f(x))=\dfrac{3}{x-1}+5 \Rightarrow \ \text{vertical translation +5 of} \ f(x)\)

\(\text{Range} \ f(x):\ \text{All} \ y, \ y \neq 0\)

\(\text{Range} \ g(f(x)): \text{All} \  y, \  y \neq 5 \ \ \text{or} \ \ \{-\infty<y<5\} \cup\{5<y<\infty\}.\)

Functions, 2ADV F1 EQ-Bank 18

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

  1. Find  \(f(g(x))\).   (1 mark)
  2. What is the domain of \(f(g(x))\) ?   (1 mark)

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 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)

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`

  1. Find  `g(f(x))`.  (1 mark)

  2. Find the domain and range of  `f(g(x))`.  (2 marks)

Show Answers Only
  1. `25 – x`
  2. `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 11

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

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

  2.  State the domain for which  `f(g(x))`  is defined.   (2 marks)

Show Answers Only
  1. `c = 2, d = 8 or c = 8, d = 2`
  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`
 

 vcaa-2011-meth-4ii

`(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`

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

  2. Let  `g(x)=x^2+4x+3`,  where  `x<=c`  and  `c<=0`.
  3. 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)

Show Answers Only
  1.   `y>= 1`
  2.  `-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 1 MC

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

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

  1. `h(x)h(–x) = –h(x^2)`
  2. `h(x) - h(0) = xh(x)`
  3. `h(x) - h(–x) = 2xh(x^2)`
  4. `(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`