Sketch the following functions/relations and supply the domain and range. (continued...)If it is a function, state whether it is into, onto or one to one from R ----> R 9x^2-18x+4y^2-8y=23

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to complete the squares `9x^2-18x` and `4y^2-8y` , hence, you need to identify the missing terms using the following formula, such that:

`(a - b)^2 = a^2 - 2ab + b^2`

If `a^2 = 9x^2` and `2ab = 18x` yields:

`a^2 = 9x^2 => a = 3x`

Replacing `3x` for a in equation `2ab = 18x` yields:

`2*3x*b = 18x => b = (18x)/(6x) => b = 3 => b^2 = 9`

Hence, you may complete the square `9x^2-18x` adding 9 both sides, such that:

`9x^2 - 18x + 9 + 4y^2 - 8y = 23 + 9`

`(3x - 3)^2 + 4y^2 - 8y = 23 + 9`

You need to complete the square `4y^2 - 8y ` , such that:

`a^2 = 4y^2 => a = 2y`

`2ab = 8y => 2*2y*b = 8y => b = (8y)/(4y) => b = 2 => b^2 = 4`

Hence, you may complete the square `4y^2 - 8y` adding 4 both sides, such that:

`(3x - 3)^2+ 4y^2 - 8y + 4 = 23 + 9 + 4`

`(3x - 3)^2+ (2y - 2)^2 = 23 + 9 + 4`

`(3x - 3)^2+ 4y^2 - 8y + 4 = 36`

`9(x - 1)^2 + 4(y - 1)^2 = 36`

Dividing by 36 yields:

`(x - 1)^2/4 + (y - 1)^2/9 = 1`

The obtained equation represents the equation of an ellipse with vertical major axis, translated with (1,1).

The domain of the ellipse function is the interval `[h-a;h+a] = [1 - 2; 1 + 2] = [-1;3]` and the range is `[k-b,k+b] = [1-3,1+3] = [-2,4].`

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