smc-4244-10-Domain

Functions and Graphs, SMB-015

State the inequality that defines the domain of the function `g(x) = 2/sqrt(5-x)` ? (2 marks)

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`text(Domain)\ g(x):\ \ x<5`

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`g(x) = 2/sqrt(5-x)\ \ text{exists for:}`

`5-x`	`> 0\ \ \ (5-x!=0)`
`-x`	`> -5`
`x`	`<5`

`:.\ text(Domain)\ g(x):\ \ x<5`

Functions and Graphs, SMB-014

What is the domain of the function `g(x) = log_2(x^2-3)`? (2 marks)

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`text{Domain:}\ x>sqrt3\ \ ∪\ \ x<-sqrt3`

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

`x^2-3`	`> 0`
`x^2`	`> 3`

`x>sqrt3\ \ or\ \ x<-sqrt3`

`:.\ text{Domain:}\ x>sqrt3\ \ ∪\ \ x<-sqrt3`

Functions and Graphs, SMB-013

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

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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`

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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-012

What is the domain of the function `f(x) = log_10(3-2x)`? (2 marks)

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`text{Domain:}\ x<3/2`

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

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

Functions and Graphs, SMB-011

What is the domain of the function `g(x) = log_2(x+1)`? (2 marks)

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`text{Domain:}\ x> -1`

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

`x+1`	`> 0`
`x`	`> -1`

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`

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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-006

State the domain of the function `f(x) = x^2 + log_10(x)`. (2 marks)

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`x>0`

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`f(x) = x^2 + log_10(x)`

`x^2 \ text(is defined for all)\ x`

`log_10 x \ text(is defined for)\ \ x > 0`

`:. \ text(Domain)\ f(x): x>0`

Functions and Graphs, SMB-005 MC

The domain of the function `f (x) = log_2 (2x + 1)` is

`-1/2<x<0`
`text{All}\ x`
`x> -1/2`
`– oo<x<-1/2`

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`C`

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`text(Domain exists for:)`

`2x + 1`	`> 0`
`2x`	`> -1`
`x`	`> -1/2`

`=> C`

Functions and Graphs, SMB-004 MC

What is the domain of the function `f(x) = log_10(4-x)`?

`x < 4`
`x <= 4`
`x > 4`
`x >= 4`

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`A`

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`4-x`	`> 0`
`-x`	`> -4`
`x`	`< 4`

`=> A`

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-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`

Functions, 2ADV F1 2020 HSC 1 MC

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

`x < 3/2`
`x > 3/2`
`x <= 3/2`
`x >= 3/2`

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`D`

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`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, 2ADV F1 2013 HSC 3 MC

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

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

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`A`

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`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`