smc-4244-20-Range

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-009

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

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

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`text{Range}\ f(x): \ y>=1`

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`text{Consider the function transformation:}`

`y=2x^2\ text{translated 1 unit up}\ \ =>\ \ y=2x^2+1`

`(x^2>0\ text{for all}\ x)`

`:.\ text{Range}\ f(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`

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`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-007

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

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

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`text{Range}\ g(x): \ y>=-1`

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`text{Consider the function transformation:}`

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

`:.\ text{Range}\ g(x): \ y>=-1`

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`

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`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`

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`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`.

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

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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`

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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`