1. Given that f(x) = x^2 − 2x and that g(x) = sqrt(x-15):

A. State (g • f)(x) and (g + f)(x).
B. Find all vertical asymptotes of (g/f)(x).
C. Determine the domain of (g ○ f)(x).
D. Determine the range of (g ○ f)(x).

(g • f)(x)

= √(x-15)(x^2-2x) , can you see what I did?

now do the same for (g+f)(x)

similarly , (g/f)(x) = √(x-15)/(x^2 - 2x)

B) for VA's the denominator is zero
so when is x^2 - 2x = 0
x(x-2) = 0
x = 0 and x = 2 are the vertical asymptotes

C)
(g ○ f)(x)
= g(f(x) )
= g(x^2-2x)
= √((x^2-2x)-15)
= √((x-5)(x+3)

domain is your choice of x's that you may use in the expression to yield a real number.
Clearly the inside of the square root has to be ≥ 0
this is true for x ≤ -3 OR x ≥ 5

D) repeat my method of C), remember that range is the resulting output from your function, that is , your y values.

No calculators are allowed for this question.

A. To find (g • f)(x), we need to first find g(f(x)):

F(x) = x^2 - 2x

g(f(x)) = sqrt(f(x) - 15)
= sqrt(x^2 - 2x - 15)

To find (g + f)(x), we add the two functions together:

(g + f)(x) = g(x) + f(x)
= sqrt(x - 15) + (x^2 - 2x)

B. To find the vertical asymptotes of (g/f)(x), we need to determine where the denominator is equal to zero. In this case, the denominator is f(x):

f(x) = x^2 - 2x

Setting f(x) equal to zero and solving for x:

x^2 - 2x = 0
x(x - 2) = 0

Therefore, the vertical asymptotes are x = 0 and x = 2.

C. To determine the domain of (g ○ f)(x), we need to find where the composition g(f(x)) is defined.

The domain of g(x) = sqrt(x - 15) is x ≥ 15.

The domain of f(x) = x^2 - 2x is all real numbers.

Since f(x) is defined for all real numbers, we need to make sure that the output of f(x) falls within the domain of g(x). Therefore, we need to find the range of f(x) and see if it falls within the domain of g(x).

D. To determine the range of (g ○ f)(x), we need to find the range of g(f(x)).

Since f(x) = x^2 - 2x, we can find its vertex to determine the minimum or maximum value. The vertex can be found using the formula -b/2a:

x = -(-2) / 2(1) = 1

Substituting x = 1 back into the equation:

f(1) = (1)^2 - 2(1) = -1

Therefore, the vertex is (1, -1).

Since a > 0, the parabola opens upward, which means the range of f(x) is all real numbers greater than or equal to -1.

Therefore, the range of (g ○ f)(x) is all real numbers greater than or equal to the minimum value of g(x), which is the square root of 15.

A. To find (g • f)(x), we need to apply the function g to the function f. This is done by substituting f(x) into g(x). So we have:

(g • f)(x) = g(f(x)) = g(x^2 - 2x)

Similarly, to find (g + f)(x), we need to add the functions g(x) and f(x) together:

(g + f)(x) = g(x) + f(x) = sqrt(x-15) + (x^2 - 2x)

B. To find the vertical asymptotes of (g/f)(x), we need to determine where the denominator, f(x), becomes zero. In this case, we need to find the values of x that satisfy the equation f(x) = 0.

x^2 - 2x = 0

Factoring out x, we have:

x(x - 2) = 0

Therefore, x = 0 or x = 2.

These are the values of x that we need to avoid when finding the domain of (g/f)(x), as they would result in division by zero.

C. To determine the domain of (g ○ f)(x), we need to consider the composition of functions g and f. Since we are working with square roots, the expression inside the square root of g(x) (which is f(x)) must be greater than or equal to zero.

x^2 - 2x ≥ 0

Factoring out x, we have:

x(x - 2) ≥ 0

This inequality is satisfied when x ≤ 0 or x ≥ 2.

Therefore, the domain of (g ○ f)(x) is x ≤ 0 or x ≥ 2.

D. To determine the range of (g ○ f)(x), we need to consider the range of the composition of functions g and f. Since g(x) = sqrt(x-15), the range of g(x) is all non-negative real numbers.

For f(x) = x^2 - 2x, we can find the minimum value of the function by completing the square. The function is a parabola opening upwards, so the minimum value is at the vertex. The vertex of the parabola is given by x = -b/2a, where a = 1 and b = -2. Therefore, the minimum value occurs at x = 1.

Substituting x = 1 into f(x), we have:

f(1) = 1^2 - 2(1) = -1

The range of f(x) is all real numbers less than or equal to -1.

Since (g ○ f)(x) is the composition of g and f, the output of (g ○ f)(x) will be all non-negative real numbers that are less than or equal to -1.

Therefore, the range of (g ○ f)(x) is all real numbers less than or equal to -1.