In the following relations, y is a function of x
[tex](a)\ (x - 6) = 3(y - 3)[/tex]
[tex](b)\ x^2 - 2x + 3y = 4[/tex]
In the following relations, y is not a function of x
[tex](c)\ y^3 - 4 =0[/tex]
[tex](d)\ 2|x| + |y| = 2[/tex]
The domain and the range of each relation will be calculated using graphs (see attachment)
[tex](a)\ (x - 6) = 3(y - 3)[/tex]
Open brackets
[tex]x - 6 = 3y - 9[/tex]
Collect like terms
[tex]3y =x - 6 + 9[/tex]
[tex]3y =x +3[/tex]
Divide both sides by 3
[tex]y = \frac 13x + 1[/tex]
We have the following observations:
- y is a function of x
- It is a one-to-one function, because every value of x, has 1 corresponding y-value
- The domain and the range are [tex](-\infty,\infty)[/tex]
[tex](b)\ x^2 - 2x + 3y = 4[/tex]
Rewrite as:
[tex]3y = -x^2 + 2x + 4[/tex]
Divide through by 3
[tex]y = \frac{1}{3}(-x^2 + 2x + 4)[/tex]
We have the following observations:
- y is a function of x
- The domain of the function is [tex](-\infty,\infty)[/tex].
- The range of the function is [tex](-\infty, \frac 53][/tex]
[tex](c)\ y^3 - 4 =0[/tex]
Rewrite as:
[tex]y^3 = 4[/tex]
We have the following observations:
- y is not a function of x,
- The domain of the function is [tex](-\infty,\infty)[/tex].
- The range of the function is [tex]\sqrt[3]4[/tex]
[tex](d)\ 2|x| + |y| = 2[/tex]
Rewrite as:
[tex]|y| = 2 - 2|x|[/tex]
We have the following observations:
- y is not a function of x
- The domain of the function is [tex][-1,1][/tex].
- The range of the function is [tex][-2,2][/tex]
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