Question 12

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

The equation x = y² + 2 does not express y purely in terms of x, which is a requirement for a relation to be classified as a function of x. In this case, for a given x value, there can be two corresponding y values (positive and negative square roots), violating the definition of a function.

A) y = x

This equation represents y directly as a function of x, where for each x value there is a unique corresponding y value. It satisfies the definition of a function since it describes a linear relationship.

B) y = x + 2

Similar to option A, this equation clearly defines y in terms of x, resulting in a unique output for each input. It is straightforward and adheres to the function definition.

C) y = x² + 2

This equation also defines y as a function of x, where each x value yields a single y value, specifically the square of x plus two. The relationship is well-defined and fits the criteria for a function.

D) x = y + 2

Although this equation is rearranged to express x in terms of y, it can be rewritten as y = x - 2, thereby allowing y to be expressed as a function of x. Thus, it does not meet the criteria for the question as it can still establish a unique y for each x.

E) x = y² + 2

This equation does not express y solely in terms of x. For a single x value, there can be two possible y values (positive and negative), which means it does not fulfill the definition of a function, as one input does not yield a unique output.

Conclusion

The equation x = y² + 2 is the only option that fails to represent y uniquely as a function of x, as it allows multiple outputs for a single input. In contrast, all other options (A, B, C, and D) successfully define y in terms of x, adhering to the fundamental definition of a function.