Final answer:
The equation y = (x³)² - 5 has an infinite number of solutions because it is a polynomial function, which is defined for all real numbers, leading to an infinite number of (x, y) pairs. Option D is the correct answer.
Step-by-step explanation:
The equation in question is y = (x³)² - 5. To find out how many solutions the equation has, we need to realize that y is defined for all values of x since the function is a polynomial, which means the domain is all real numbers. This type of equation represents a sixth-degree polynomial, as the exponent on x after expanding would be six. For any real value of x that we plug into this function, we will get a corresponding y-value.
This is because polynomials of degree higher than zero are continuous functions and will always produce some y-value for every x in their domain. So, the only way the function could fail to have a solution is if there were some restrictions on the domain of x or codomain of y, which there is not in this case.
When looking for the number of solutions, we are generally interested in the number of x-values that can make the equation equal to zero (the number of roots), but in the context of this function, since there are no restrictions, there is an infinite number of (x, y) pairs that satisfy the equation for various values of x. Therefore, the correct answer is D) Infinite.