Question

Determine whether the equation defines y as a function of x.
x+y=9 and x^2+y^2=1 and x=y^2

Answer 1

Function:
`x+y=9`

Not Function:
`x^2+y^2=1` and
`x=y^2`

Explanation:

Given

`x+y=9`

`x^2+y^2=1`

`x=y^2`

Required

Determine if
`y` is a function of
`x`

Solving x+y=9

`x+y=9`

Make y the subject of formula

`y = 9 - x`

Hence; y is a function of x

Solving
`x^2+y^2=1`

`x^2+y^2=1`

Subtract x² from both sides

`y^2=1 - x^2`

Square root of both sides

`y =\± √(1 - x^2)`

This implies that

`y =√(1 - x^2)` or
`y =-√(1 - x^2)`

Because
`y` can be any of those two expressions, it is not a function.

Solving
`x=y^2`

`x=y^2`

Reorder

`y^2 = x`

Take square roots

`y = \±√(x)`

This implies that

`y = √(x)` or
`y = -√(x)`

Because
`y` can be any of those two expressions, it is not a function.