determine whether 14y3-14-12x2=3x defines y as a function of x. Please show all of your work.Need correct answer
You need to start by isolating the term containing y to the left side such that:
`14y^3 = 14 + 12x^2 + 3x`
You need to divide by 14 both sides:
`y^3 = (14 + 12x^2 + 3x)/14`
You need to take cube root both sides such that:
`root(3)(y^3) = root(3) ((14 + 12x^2 + 3x)/14)`
`y = root(3) ((14 + 12x^2 + 3x)/14)`
Since, the cube root imposes no limitations for values of x, hence y is defined as a function of x such that: `y = root(3) ((14 + 12x^2 + 3x)/14).`
A function y = f(x) is defined if for any value of x there is only one value of y. The relation `y = sqrt x` is not a function of x as for any value of x there are two values of y as the square of `+sqrt x` and `-sqrt x` is x.
The given relation 14y^3-14-12x^2=3x defines can be written as a function of x if each value of x corresponds to only one value of y.
14y^3 = 3x + 12x^2 + 14
y^3 = `(3x + 12x^2 + 14)/14`
y = `root(3)((3x + 12x^2 + 14)/14)`
This is a function as the cube root of a negative number is negative and that of a positive number is positive. For no value of x do we get two values of y.