Final answer:
Option C, y – x² = 4 or rearranged as y = x² + 4, is the correct answer as it is the only equation where each x-value corresponds to exactly one y-value, which satisfies the definition of a function.
Step-by-step explanation:
To determine which of the given equations represents a function, we must apply the vertical line test theoretically. A graph represents a function if and only if no vertical line intersects the graph at more than one point. When we look at the options:
- A. y² + x = 6
- B. y´ + x³ = 3
- C. y – x² = 4
- D. y² – x² = 7
- E. y² + x² = 7
Options A, B, D, and E cannot represent functions because, for any given value of x, there could be more than one corresponding value of y. This happens because the equation involves y squared or raised to the fourth power, indicating that y can take multiple values for a single x (positive and negative roots).
Option C, y – x² = 4, is the equation that represents a function. This can be rearranged to y = x² + 4, which is a parabola opening upwards. For each value of x, there is only one corresponding y value, meaning it passes the vertical line test and thus, is a function.