The domain of an expression is the set of input values the expression can take
- The domain of the expression is [tex]y \ne 1.5x[/tex]
- The domain of the simplified expression is [tex]y \ne -1.5x[/tex]
(a) The domain of the expression
The expression is given as:
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)}[/tex]
Set the denominator to 0, to determine the domain
[tex](4y^2 - 9x^2) = 0[/tex]
Remove brackets
[tex]4y^2 - 9x^2 = 0[/tex]
Add 9x^2 to both sides
[tex]4y^2 = 9x^2[/tex]
Divide both sides by 4
[tex]y^2 = 2.25x^2[/tex]
Take positive square roots of both sides
[tex]y = 1.5x[/tex]
Hence, the domain of the expression is [tex]y \ne 1.5x[/tex]
(b) The domain of the simplified expression
The expression is given as:
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)}[/tex]
Rewrite as:
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{(9x^2 - 12xy+ 4y^2 )}{((2y)^2 - (3x)^2)}[/tex]
Expand the numerator
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{(9x^2 - 6xy - 6xy+ 4y^2 )}{((2y)^2 - (3x)^2)}[/tex]
Factorize the numerator
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{3x(3x - 2y) - 2y(3x- 2y )}{((2y)^2 - (3x)^2)}[/tex]
Factor out 3x - 2y
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{(3x - 2y)(3x- 2y )}{((2y)^2 - (3x)^2)}[/tex]
Express the denominator as a difference of two squares
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{(3x - 2y)(3x- 2y )}{(2y - 3x)(2y + 3x))}[/tex]
Rewrite as:
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{-(2y - 3x)(3x- 2y )}{(2y - 3x)(2y + 3x))}[/tex]
Cancel out the common term
[tex]\frac{(9x^2 + 4y^2 - 12xy)}{(4y^2 - 9x^2)} = \frac{-(3x- 2y )}{(2y + 3x)}[/tex]
Set the denominator to 0, to determine the domain
[tex]2y + 3x= 0[/tex]
Subtract 3x from both sides
[tex]2y= -3x[/tex]
Divide both sides by 2
[tex]y= -1.5x[/tex]
Hence, the domain of the simplified expression is [tex]y \ne -1.5x[/tex]
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