Question

How do I solve this function question?

Answer 1

y is a function of x if every value of x in the domain maps to exactly one value of y in the range. If you can find a single value of x that maps to two or more values of y, then y is not a function of x.

To figure out if this is possible, suppose that there are two different values of y (which we’ll call y₁ and y₂) that result in the same x. In other words, assume you have y₁ and y₂ such that:
> y₁ ≠ y₂ ; and

> (3y₁ + 5)/(7y₁ - 1) = (3y₂ + 5)/(7y₂ - 1)

Solve this equation, and see if y₁ = y₂. If it does, then this contradicts the assumption that y₁ ≠ y₂, and you can conclude that y *is* a function of x. If you find that y₁ ≠ y₂ holds true, then that shows that y is *not* a function of x.

(3y₁ + 5)/(7y₁ - 1) = (3y₂ + 5)/(7y₂ - 1)
> (3y₁ + 5)(7y₂ - 1) = (3y₂ + 5)(7y₁ - 1)

> 21y₁y₂ - 3y₁ + 35y₂ - 5 = 21y₁y₂ - 3y₂ + 35y₁ - 5

> 21y₁y₂-21y₁y₂ - 5 + 5 = -3y₂ + 3y₁ + 35y₁ - 35y₂

> 0 = -3y₂ + 3y₁ + 35y₁ - 35y₂

> 35y₂ + 3y₂ = 35y₁ + 3y₁

> 38y₂ = 38y₁

> y₂ = y₁

This is a contradiction, so y is a function of x.

Answer 2

For each x is there a unique y satisfying the equation?

Hint: try solving for y.

Answer 3

If you have learned the parts, then you know what it's asking.

It's asking if y is a function. You've learned the vertical line test?

Of course, you have to know what y is, so solve for it.