The graph of the function f(x) = (x − 3)(x + 1) is shown. which describes all of the values for which the graph is positive and decreasing?

All real values of x where x < −1

Step-by-step explanation:

Solve the function as an inequality:

(x-3)(x+1)>0

There are two cases in this:

Both factors are greater than 0.

x-3>0 and x+1>0

x>3 and x>-1

The intersection is x >3

OR

Both factors less than 0.

x-3<0 and x+1<0

x< 3 and x<-1

The intersection is x< -1

We have obtained that the function is positive for the intervals  x < -1 and x > 3. But in one of those intervals the function is decreasing and in the other is increasing.

The answer is: all real values of x where x < −1

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).