Given the following function f(x)=-x^2-10x. Find the domain, range and decreasing intervals?

as with all polynomials, the domain is all real numbers.

f(x) = -x(x+10)
This is a parabola opening downwards. Its maximum occurs midway between its root, at x = -5
f(-5) = 25
so, the range is y <= 25

Since its maximum is at x = -5, it decreases for x in (-5,oo).

Well, let's analyze the function f(x)=-x^2-10x.

The domain of a function refers to all the possible x-values for which the function is defined. Since there are no restrictions or excluded values for x, the domain of f(x) is all real numbers, or (-∞, ∞).

To determine the range of the function, we need to find the set of all possible y-values or outputs. By observing the graph of f(x)=-x^2-10x, we notice that the quadratic term (-x^2) has a negative coefficient, which means the parabola opens downwards. As such, the vertex represents the maximum point of the function. In this case, the vertex occurs when x = -5 (obtained by using the formula -b/(2a), where a and b are coefficients of the quadratic). Substituting x = -5 into f(x), we get f(-5) = -5^2 - 10(-5) = -25 + 50 = 25. Hence, 25 is the maximum value of the function.

Therefore, the range of f(x) is (-∞, 25]. In other words, f(x) can take any y-value less than or equal to 25.

As for finding the decreasing intervals, we need to identify where the function is decreasing or getting smaller. Since f(x) is a quadratic with a negative leading coefficient (-1), it is decreasing on the interval (-∞, -5] and [∞, -5). This means that as x moves towards -5 from either the left or right, the function values decrease.

So, the domain of f(x) is (-∞, ∞), the range is (-∞, 25], and the decreasing intervals are (-∞, -5] and [∞, -5).

To find the domain of a function, we need to determine all possible values of x for which the function is defined. In this case, we don't have any restrictions on x, so the domain is all real numbers (-∞, ∞).

To find the range of a function, we need to determine all possible values of f(x) for the given domain. In this case, the function is a quadratic function in the form f(x) = -x^2 - 10x. The coefficient of x^2 is negative, which means the parabola opens downward. The maximum value of the function occurs at the vertex.

To find the vertex of a quadratic function in the form f(x) = ax^2 + bx + c, we can use the formula:

x = -b / (2a)
f(x) = f(-b / (2a))

In this case, we have a = -1, b = -10, and c = 0:

x = -(-10) / (2*-1) = 10 / -2 = -5
f(-5) = -(-5)^2 - 10(-5) = -25 + 50 = 25

So the vertex of the parabola is at (-5, 25). Since the parabola opens downward, the maximum value is 25, and the range of the function is (-∞, 25].

To find the decreasing intervals, we need to determine the x-values for which the function is decreasing. Since the function is a quadratic function with a negative coefficient for the x^2 term, the function is always decreasing. So the decreasing intervals cover the entire domain (-∞, ∞).

To find the domain, range, and decreasing intervals of the function f(x) = -x^2 - 10x, we'll consider each one separately.

First, let's find the domain of the function. The domain represents all possible input values of x for which the function is defined. In this case, since the function is a polynomial, it is defined for all real numbers. Therefore, the domain of f(x) is (-∞, ∞).

Next, let's determine the range of the function. The range represents all the output values that the function can produce. To find the range, we need to analyze the behavior of the function. The leading coefficient of the quadratic term (-x^2) is negative, indicating that the graph opens downwards. Additionally, there is no other term that can offset the negative square term. Therefore, the maximum value of the function is obtained at the vertex. To find the vertex, we can use the formula x = -b/2a, where a is the coefficient of the square term (-1 in this case), and b is the coefficient of the linear term (-10 in this case).

x = -(-10)/(2*(-1))
x = 5

To find the corresponding y-value (f(x)) at the vertex, substitute x = 5 into the function:
f(5) = -(5^2) - 10(5)
f(5) = -25 - 50
f(5) = -75

Hence, the vertex of the parabola is (5, -75).

Since the parabola opens downwards, the range is from the vertex's y-value (-75) to negative infinity. Therefore, the range of f(x) is (-∞, -75].

Finally, let's identify the decreasing intervals of the function. In this case, since the leading coefficient is negative, the function is decreasing on the intervals where it is defined. To find these intervals, we can examine the sign of the derivative, which gives information about the slope of the function. The derivative of f(x) = -x^2 - 10x is:

f'(x) = -2x - 10

To find where the derivative is negative (indicating a decreasing interval), we set f'(x) < 0 and solve for x:

-2x - 10 < 0
-2x < 10
x > -5

Therefore, the function is decreasing for all x values greater than -5. In interval notation, the decreasing interval is (-5, ∞).