Hello, could someone check my answers to the following
1.For the function f(x) = x^2 - 2x + 1,
(a) find f(0)
My Ans:
(0) = 0² - 2(0) + 1
= 0 - 0 + 1
= 1
f(0) = 1
(b) solve f(x) = 0
My Ans:
0 = x² - 2x + 1
(x - 1)(x - 1) = 0
(x - 1)² = 0
x = 1
2. (Solve and check) The amount of money that a salesperson makes varies directly as the total amount of sales made. If the salesperson receives $7,500 for selling a house for $120,000, how much will he or she make if he or she sells a house for $150,000?
My Ans:
x = commission on $150,000
120,000/ 150,000 = 7500/x
120,000x = 150,000 * 7500
120,000x = 1125000000
x = 1125000000/120,000
x = $9,375 (how do I do the check?)
3. If y varies inversely as x^2, and y = 2 when x = 4, what is the constant of variation?
My Ans:
y = k/x^2
2 = k/4^2
2 = k/16
k = 32
I not very sure with this, especially #3.
Thanks in advance.
good job, all answers are correct.
for a check of #2 you could find the rate of commission
7500/120000 = .0625
then .0625(150000) = 9375
1. For the function f(x) = x^2 - 2x + 1,
(a) To find f(0), you substitute x = 0 into the function:
f(0) = (0)^2 - 2(0) + 1
= 0 - 0 + 1
= 1
Your answer of f(0) = 1 is correct.
(b) To solve f(x) = 0, you set the function equal to zero and solve for x:
0 = x^2 - 2x + 1
To factor this quadratic, you notice that it's a perfect square trinomial:
0 = (x - 1)^2
Setting (x - 1)^2 = 0, you can solve for x:
x - 1 = 0
x = 1
Your answer of x = 1 is correct.
2. For the salesperson's commission problem, you correctly set up the proportion using direct variation:
120,000 / 150,000 = 7,500 / x
To solve for x, you cross-multiply and solve the resulting equation:
120,000x = 150,000 * 7,500
120,000x = 1,125,000,000
x = 1,125,000,000 / 120,000
x = $9,375
To check your answer, you substitute x = $9,375 back into the original proportion:
120,000 / 150,000 = 7,500 / 9,375
Dividing both sides by 30,000 simplifies the equation:
0.8 = 0.8
The two sides are equal, so your answer of x = $9,375 is correct.
3. For the inverse variation problem, you have the equation y = k / x^2. Given that y = 2 when x = 4, you can substitute these values into the equation to find the constant of variation, k:
2 = k / (4^2)
2 = k / 16
To solve for k, you multiply both sides by 16:
32 = k
So the constant of variation, k, is 32.
Your answer for #3 is correct.