Question

What is the range of the function y = -x2 + 1?

Answer 1

The range of the function y = -x^2 + 1 is all real numbers less than or equal to 1, represented as ]-∞, 1].

Step-by-step explanation:

The range of the function y = -x^2 + 1 refers to the set of all possible values that y can take as x varies over all real numbers. Since the function is a downward-opening parabola (due to the negative coefficient of x^2), the maximum value of y is at the vertex of the parabola. The vertex, in this case, occurs at x=0, which gives us the maximum y value of y = 1. As x moves away from zero in either direction, the value of y decreases and approaches negative infinity. Hence, the range of this function is ]-∞, 1], or all real numbers less than or equal to 1.

Answer 2

To solve this problem you must apply the proccedure shown below:

1- You have the following function given in the problem above:

y=-x² + 1

2- If you graph the function, you will obtain the parabola shown in the figure attached. As you can see, the vertex is ( the maximum value of y is y coordinate of the vertex is (0,1) and the parabola opens down. Therefore the range is:

Range: {y∈R:y≤1}

Therefore, the answer is: {y∈R:y≤1}