Which of the following equations does not represent y as a function of x in the standard (x, y) coordinate plane?

Rationale

x = y² + 2 does not represent y as a function of x.

This equation implies that for a given value of x, there can be multiple corresponding values of y, specifically two values (both the positive and negative square roots) for most x-values. Therefore, it violates the definition of a function, which states that each input must have exactly one output.

A (y = x) YOUR ANSWER

This equation represents a linear function where each value of x corresponds directly to one unique value of y. It is a classic example of a function, as it satisfies the criteria of having a single output for every input.

B (y = x + 2) YOUR ANSWER

Similar to option A, this is also a linear function. For any input value of x, there is a unique output value of y, fulfilling the definition of a function. This equation shifts the line y = x upward by 2 units.

C (y = x² + 2) YOUR ANSWER

This quadratic function maintains the characteristic of a function as well, since for each x-value, there is exactly one corresponding y-value. The graph is a parabola that opens upwards, reflecting unique outputs for each input.

D (x = y + 2) YOUR ANSWER

Although this equation is presented in a non-standard form, it can be rearranged to y = x - 2, which is still a linear function. Each value of x will yield a single corresponding value of y, thus satisfying the function requirement.

Conclusion

In summary, while options A, B, C, and D all represent valid functions where each input x produces a unique output y, option E, x = y² + 2, fails this test. It allows multiple values of y for the same x, violating the fundamental definition of a function. Understanding these distinctions is crucial for proper function analysis in mathematics.