1. The range of f ( x ) = ax + b is the set of all real numbers given that a and b are real numbers. Which produces a counter examples to her statement ?
A. a = 0
B. b = 0
C. a < 0
D. b < 0
My answer I came up with was b or c I was not sure.
2. f ( x ) = - 6
x -1 and g ( x ) 4 x^2
select the solution for ( f +g ) ( x )
A. - 6 x - 1 + 4 x ^ 2
B. - 24 x^3 - 4 x^2
C. 24 x^2 - 1
D. - 6 x - 1 24 x^2
My answer I came up with is C.
3. f ( x ) = - 6 x - 1 and g ( x ) = 4 x^2 select the solution for ( f g )( x )
A. - 6 x - 1 + 4 x^2
B. -24 x^3 -4 x^2
C. - 24 x^2 - 1
D . -24 x^5
My answer I came up with was either C or D. was not sure.
4. The profit ( p) , in dollars , for a company is modeled by the function P ( x ) = - 750 x^2 + 15,000 x , Where x is the number of items produced. For which x is the number of items produced. For which values of x will the company lose money ?
A. x > 20
B. 2 < x < 10
C. 10 < x < 20
D. x < 2
My answer I came up with was A.
5. The function h ( x ) = 1 / 98 x^2 describes h ( x ) , the height of part of a roller coaster track , where x is the horizontal distance in feet from the center of this section of the track, The towers that support this part of the track are the same height and are 150 feet apart, which is the best estimate of the height of the towers ?
A. 121.2 feet
B. 85. 7 feet
C. 229. 6 feet
D. 57.4 feet
My answer I came up with was C. I am not sure .
#1 is (A) because then f(x) = 0x+b = b
and the range is just y=b
#2 (f+g)(x) = f(x) + g(x). So, (A)
#3 (fg)(x) = f(x) * g(x) = (-6x-1)(4x^2). So, (B)
#4 P(x) = - 750 x^2 + 15,000 x = 750x(20-x)
This is a parabola with roots at x=0,20
Since it opens downward, P(x) is positive only between the roots.
So, (A)
#5 Since the poles are 150 apart, -75<=x<=75
h(75) = 1/98 x^2 = 1/98 * 75^2 = 57.4
So, (D)
Thank you Stevie
find the function h(x)=1/98 x^2
1. To determine which produces a counter example to her statement, we need to find a value of a or b that would make the range of f(x) not the set of all real numbers.
The range of f(x) = ax + b can be any real number because a and b are both real numbers, and for any x value, the expression ax + b will produce a real number.
A. a = 0: If a = 0, then the function simplifies to f(x) = b. The range of f(x) would then only be a single value, b, and not all real numbers. Therefore, A is a counter example.
B. b = 0: If b = 0, then the function simplifies to f(x) = ax. Since a can still take any real value, the range of f(x) would still be all real numbers. Therefore, B is not a counter example.
C. a < 0: If a is negative, the function will produce negative values for all x values. However, the range of f(x) would still be all real numbers because negative numbers are real numbers. Therefore, C is not a counter example.
D. b < 0: If b is negative, the function will shift down and produce negative values, but the range of f(x) would still include all real numbers. Therefore, D is not a counter example.
Based on this analysis, the counter example to her statement is A. a = 0.
2. To find the solution for (f + g)(x), we need to add the functions f(x) and g(x).
f(x) = -6x - 1
g(x) = 4x^2
Adding these two functions, we get:
(f + g)(x) = (-6x - 1) + (4x^2)
Simplifying, we get:
(f + g)(x) = 4x^2 - 6x - 1
Therefore, the solution for (f + g)(x) is B. - 24x^3 - 4x^2.
3. To find the solution for (fg)(x), we need to multiply the functions f(x) and g(x).
f(x) = -6x - 1
g(x) = 4x^2
Multiplying these two functions, we get:
(fg)(x) = (-6x - 1)(4x^2)
Simplifying, we get:
(fg)(x) = -24x^3 - 4x^2
Therefore, the solution for (fg)(x) is B. -24x^3 - 4x^2.
4. To find the values of x for which the company will lose money, we need to determine when the profit (P(x)) is negative.
The profit for the company can be modeled by the function P(x) = -750x^2 + 15,000x.
To find when the company loses money, we need to find when the profit (P(x)) is negative. In other words, we need to find the values of x for which P(x) < 0.
Setting P(x) < 0, we have:
-750x^2 + 15,000x < 0
Factoring out common factors, we get:
-750x(x - 20) < 0
To find the values of x that satisfy the inequality, we can use a sign chart or test intervals. The critical points are x = 0 and x = 20.
Testing the intervals:
For x < 0, we choose x = -1: -750(-1)(-1 - 20) > 0, which is false.
For 0 < x < 20, we choose x = 1: -750(1)(1 - 20) < 0, which is true.
For x > 20, we choose x = 21: -750(21)(21 - 20) > 0, which is false.
Therefore, the values of x for which the company will lose money are 0 < x < 20. Thus, the answer is C. 10 < x < 20.
5. The function h(x) = 1/98 x^2 describes the height of a part of a roller coaster track as a function of the horizontal distance from the center of that section of the track.
To find the best estimate of the height of the towers, we need to evaluate the function at the midpoint between two towers. The given information states that the towers are 150 feet apart.
The midpoint between two towers is halfway between them, which is 150/2 = 75 feet away from the center of each tower.
Plugging this value into the function, we have:
h(75) = 1/98 * 75^2
Simplifying, we get:
h(75) = 1/98 * 5625
Calculating, we find:
h(75) ≈ 57.4 feet
Therefore, the best estimate of the height of the towers is D. 57.4 feet.