To determine the domain and range of the quadratic function [tex]\( f(x) = -x^2 \)[/tex], follow these steps:
### Domain:
The domain of a function refers to all the possible input values (x-values) that the function can accept.
In the case of the quadratic function [tex]\( f(x) = -x^2 \)[/tex]:
1. Identify the limitations on x: For a quadratic equation, there are no restrictions on the values that x can take. It can be any real number since x can be positive, negative, or zero.
Therefore, the domain of the function [tex]\( f(x) = -x^2 \)[/tex] is all real numbers, which can be written as:
[tex]\[ \text{Domain:} \quad (-\infty, \infty) \][/tex]
### Range:
The range of a function is the set of all possible output values (y-values) that the function can produce.
To find the range of the function [tex]\( f(x) = -x^2 \)[/tex]:
1. Determine the behavior of the quadratic function: Since the coefficient of [tex]\( x^2 \)[/tex] is negative (a = -1), the parabola opens downward. This means the function will have a maximum value and will decrease as you move away from the vertex in either direction.
2. Find the vertex of the function: The vertex form of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] helps in finding the maximum or minimum. In this case, the function has a vertex at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Since [tex]\( b = 0 \)[/tex] in [tex]\( f(x) = -x^2 \)[/tex], the vertex is at:
[tex]\[ x = -\frac{0}{2(-1)} = 0 \][/tex]
3. Calculate the y-value at the vertex: Plug the x-value of the vertex back into the function to determine the maximum y-value:
[tex]\[ f(0) = -0^2 = 0 \][/tex]
4. Determine the range based on the maximum value: Since the parabola opens downward and the maximum value is 0, the range includes all y-values less than or equal to this maximum value.
Therefore, the range of the function [tex]\( f(x) = -x^2 \)[/tex] is:
[tex]\[ \text{Range:} \quad (-\infty, 0] \][/tex]
### Conclusion:
For the quadratic function [tex]\( f(x) = -x^2 \)[/tex]:
- Domain: [tex]\(\text{all real numbers}\)[/tex]
- Range: [tex]\((- \infty, 0]\)[/tex]
