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Functions and Graphs, SMB-020

A circle has centre `(5,3)` and radius 3.

  1.  Describe, with inequalities, the region that consists of the interior of the circle and more than 2 units above the `x`-axis.  (2 marks)

  2.  Sketch the region.  (1 mark)

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i.    `(x-5)^2 + (y-3)^2 < 9\ ∩\ y > 2`

ii.    
       

Show Worked Solution

i.   `text(Equation of circle:)`

`(x-5)^2 + (y-3)^2 = 3^2`
 

`:.\ text(Region is:)`

`(x-5)^2 + (y- 3)^2 < 9\ ∩\ y > 2`

COMMENT: The broken line on the graph represents an excluded boundary.

 

ii.   

Functions and Graphs, SMB-019

Shade the region defined by  `y+3x>3`  on the graph below and verify your result.  (3 marks)

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`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(Incorrect – not in shaded area)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`0+3x=3\ \ =>\ \ x=1`

`ytext{-intercept occurs when}\ x=0:`

`y+3(0)=3\ \ =>\ \ y=3`

`text{Test}\ (0,0):`

`0+3(0)>3\ \ =>\ \ 0>3\ \ text{(Incorrect – not in shaded area)}`

Functions and Graphs, SMB-018

Shade the region defined by  `x/2-y>0`  on the graph below and verify your result.  (2 marks)

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`text{Test}\ (1,0):`

`1/2-0>0\ \ text{(correct)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`x/2-0=0\ \ =>\ \ x=0`

`text{Graph cuts at}\ (0,0)`

`text{Also passes through}\ (2,1)`
 

`text{Test}\ (1,0):`

`1/2-0>0\ \ text{(Correct – in shaded area)}`

Functions and Graphs, SMB-017

  1. Express the function  `5y-3x=15`  in the form  `y=mx+b`.  (1 mark)

  2. Shade the region defined by  `5y-3x>15`  on the graph below and verify your result.  (3 marks)
      

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i.    `y=3/5x+3`

ii.  

`text{Test}\ (0,0):`

`5(0)-3(0)>15\ \ =>\ \ 0<15\ \ text{(incorrect)}`

Show Worked Solution
i.     `5y-3x` `=15`
  `5y` `=3x+15`
  `y` `=3/5x+3`

 

ii.   `xtext{-intercept occurs when}\ y=0:`

`5(0)-3x=15\ \ =>\ \ x=-5`

`ytext{-intercept at}\ \ y=3`

`text{Test}\ (0,0):`

`5(0)-3(0)>15\ \ =>\ \ 0>15\ \ text{(Incorrect – not in shaded area.)}`

Functions and Graphs, SMB-016

Shade the region defined by  `3x-4y<12`  on the graph below and verify your result.  (3 marks)
 


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`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(correct)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`3x-4(0)=12\ \ =>\ \ x=4`

`ytext{-intercept occurs when}\ x=0:`

`3(0)-4y=12\ \ =>\ \ y=-3`

`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(Correct – is in shaded area)}`

Functions and Graphs, SMB-013

  1. Factorise the expression  `x^2-x-6`.  (1 mark)

  2. What is the domain of the function  `f(x) = log_2(x^2-x-6)`?  (2 marks)

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i.    `x^2-x-6=(x-3)(x+2)`

ii.   `text{Domain:}\ x< -2 \ ∪ \ x>3`

Show Worked Solution

i.    `x^2-x-6=(x-3)(x+2)`
 

ii.    `f(x) = log_2(x^2-x-6)=log_2(x-3)(x+2)`

`f(x)\ text{exists when:}`

`(x-3)(x+2)>0`

`x< -2 and x>3`

`text{Domain:}\ x< -2 \ ∪ \ x>3`

Functions and Graphs, SMB-010

A function has the equation  `h(x)=-1-(x-3)^2`.

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

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`text{Domain}\ h(x):\ text{all}\ x`

`text{Range}\ h(x):\ y<=-1`

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`h(x)\ text{exists for all}\ x`

`text{Domain}\ h(x):\ text{all}\ x`
 

`text{Consider the function transformation:}`

`y=x^2\ text{translated 3 units right}\ \ =>\ \ y=(x-3)^2`

`y=(x-3)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x-3)^2`

`y=-(x-3)^2\ text{translated 1 unit down}\ =>\ \ y=-1-(x-3)^2`

`:.\ text{Range}\ h(x): \ y<=-1`

Functions and Graphs, SMB-008

A function has the equation  `f(x)=4-(x+1)^2`.

State the domain and range of `f(x)`.   (3 marks)

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`text{Domain}\ f(x): \ text{all}\ x`

`text{Range}\ f(x): \ y<=4`

Show Worked Solution

`f(x)\ text{exists for all}\ x`

`text{Domain}\ f(x): \ text{all}\ x`
 

`text{Consider the function transformation:}`

`y=x^2\ text{translated 1 unit left}\ =>\ \ y=(x+1)^2`

`y=(x+1)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x+1)^2`

`y=-(x+1)^2\ text{translated 4 units up}\ \ =>\ \ y=4-(x+1)^2`

`:.\ text{Range}\ f(x): \ y<=4`

Functions and Graphs, SMB-003

State the domain and range of  `y = -sqrt(12-x^2)`.   (2 marks)

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`text(Domain:)\ -sqrt12<=x<= sqrt12`

`text(Range:)\ -sqrt12<=y<= 0`

Show Worked Solution

`y = -sqrt(12-x^2)`

`12-x^2>=0\ \ =>\ \ x^2<=12`

`:.\ text(Domain:)\ -sqrt12<=x<= sqrt12`
 

`y_max =0`

`y_min = -sqrt12\ \ (text{when}\ x=0)`

`:.\ text(Range:)\ -sqrt12<=y<= 0`

Functions and Graphs, SMB-002

A function has the equation  `f(x)=(x-2)^2-5`.

State the range of `f(x)`.   (2 marks)

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`text{Range}\ f(x): \ y>=-5`

Show Worked Solution

`text{Consider the function transformation:}`

`y=x^2\ \ text{translated 2 units to the right}\ \ =>\ \ y=(x-2)^2`

`y=(x-2)^2\ text{translated 5 units down}\ \ =>\ \ y=(x-2)^2-5`

`:.\ text{Range}\ f(x): \ y>=-5`

Functions and Graphs, SMB-001

`f(x)` is defined by the equation  `f(x)=3-x^2`.

  1. Find the coordinates of the `x`-intercepts of `f(x)`.  (1 mark)

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

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i.    `(sqrt3,0) and (-sqrt3, 0)`

ii.    `text{Domain: all}\ x`

`text{Range}\ f(x): \ y<=3`

Show Worked Solution

i.    `xtext{-intercepts occur when}\ y=0`

`3-x^2` `=0`  
`x^2` `=3`  
`x` `=+-sqrt3`  

 
`:. xtext{-intercepts at} (sqrt3,0) and (-sqrt3, 0)`
 

ii.   `text{Domain: all}\ x`

`text{Find range}\ f(x):`

`x^2>=0\ text{for all}\ x \ \ => \ \ 3-x^2<=3\ text{for all}\ x`

`:.\ text{Range}\ f(x): \ y<=3`

Functions, 2ADV F1 2020 HSC 1 MC

Which inequality gives the domain of  `y = sqrt(2x-3)`?

  1. `x < 3/2`
  2. `x > 3/2`
  3. `x <= 3/2`
  4. `x >= 3/2`
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`D`

Show Worked Solution

`text(Domain exists when:)`

`2x-3` `>= 0`
`2x` `>= 3`
`x` `>= 3/2`

  
`=>D`

Functions, 2ADV F1 2017 HSC 11h

Find the domain of the function  `f(x) = sqrt (3-x)`.  (2 marks)

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`x <= 3 or (-oo,3].`

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`text(Domain of)\ \ f(x) = sqrt (3-x)`

`3-x` `>= 0`
`x` `<= 3`

 

`text(Note domain can also be expressed as:)\ \ (-oo,3]`

Functions, EXT1* F1 2017 HSC 8 MC

The region enclosed by  `y = 4 - x,\ \ y = x`  and  `y = 2x + 1`  is shaded in the diagram below.
 

Which of the following defines the shaded region?

A.   `y <= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
B.   `y >= 2x + 1, qquad` `y <= 4-x, qquad` `y >= x`
C.   `y <= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
D.   `y >= 2x + 1, qquad` `y >= 4-x, qquad` `y >= x`
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`A`

Show Worked Solution

`text(Consider)\ \ y = 2x + 1,`

`text(Shading is below graph)`

`=> y <= 2x + 1`

`text(Consider)\ \ y = 4-x,`

`text(Shading is below graph)`

`=> y <= 4-x`

`=>  A`

Functions, EXT1* F1 2009 HSC 3c

Shade the region in the plane defined by  `y >= 0`  and  `y <= 4-x^2`.   (2 marks)

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`text(Shaded area is region where)`

`y >= 0\ text(and)\ y >= 4-x^2`

Show Worked Solution

COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(Shaded area is region where)`

`y >= 0\ \ text(and)\ \ y >= 4-x^2`

Functions, 2ADV F1 2010 HSC 1g

Let  `f(x) = sqrt(x-8)`.  What is the domain of  `f(x)`?   (1 mark)

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 `x >= 8`

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♦ Mean mark 49%.
MARKER’S COMMENT: `x>8` was a common incorrect answer.

`f(x) = sqrt(x-8)`

`text(Domain exists for:)`

`(x-8)` `>= 0`
`x` `>= 8`

Functions, EXT1* F1 2012 HSC 8 MC

The diagram shows the region enclosed by  `y = x- 2`  and  `y^2 = 4-x`. 
  

Which of the following pairs of inequalities describes the shaded region in the diagram? 

  1. `y^2 <= 4-x\ \ and\ \ y <= x-2`  
  2. `y^2 <= 4-x\ \ and\ \ y >= x-2`  
  3. `y^2 >= 4-x\ \ and\ \ y<= x-2`  
  4. `y^2 >= 4-x\ \ and\ \ y >= x-2`
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`A`

Show Worked Solution
♦  Mean mark 44%.

`text(Using information from diagram)`

`(3,0)\ text(is in the shaded region)`

`text{Substituting (3,0) into}\ \ \ y^2<=4-x,\ \ \ 0 <= 4-3 => text(true)`

`:.\ text(Cannot be)\ C\ text(or)\ D`
 

`text(Similarly)`

`(3,0)\ text(must satisfy other inequality)`

`text(i.e.)\ \ y <= x-2\ \ text(becomes)\ \ 0<= 3-2 =>\ text(true)`

`=>  A`

Functions, EXT1* F1 2013 HSC 11g

Sketch the region defined by  `(x-2)^2 + ( y-3)^2 >= 4`.    (3 marks)

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Show Worked Solution

`text(The region is the exterior of a circle,)`

COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(centre)\ text{(2,3)}\ text(and radius 2.)`
 

Functions, 2ADV F1 2013 HSC 3 MC

Which inequality defines the domain of the function  `f(x) = 1/sqrt(x+3)` ?

  1. `x > -3`  
  2. `x >= -3`  
  3. `x < -3`  
  4. `x <= -3` 
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`A`

Show Worked Solution

`text(Given)\ f(x) = 1/sqrt(x+3)`

`(x + 3)` `> 0`
`x` `> -3`

 

`:.\ text(The domain of)\ f(x)\ text(is)\ \ \ f(x)> -3`

`=>  A`