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A)(3 points)

Use the function f(x)=2x3−3x2+7
to complete the exercises.

f(−1)=


f(1)=


f(2)=


Question 2
A)
The function f(x)=12x+32
is used to complete this table.

x f(x)
-1 1
0 32
1 2
2 52
Which statements are true given the function? Select the two that apply.

(2 points)
Responses

f(1)=−1
f of 1 is equal to negative 1

f(0)=32
f of 0 is equal to 3 halves

f(2)=1
f of 2 is equal to 1

f(−12)=−2
f of open paren negative 1 half close paren is equal to negative 2

f(4)=72
f of 4 is equal to 7 halves
Question 3
A)The function p(x)=−2(x−9)2+100
is used to determine the profit on t-shirts sold for x dollars. What would the profit from sales be if the price of the t-shirts were $15 each?(1 point)
Responses

$244
$244

$172
$172

$15
$15

$28
$28
Question

1 answer

Let's work through each question step-by-step.

Question 1

Given the function \( f(x) = 2x^3 - 3x^2 + 7 \):

  1. Calculate \( f(-1) \): \[ f(-1) = 2(-1)^3 - 3(-1)^2 + 7 \] \[ = 2(-1) - 3(1) + 7 \] \[ = -2 - 3 + 7 = 2 \]

  2. Calculate \( f(1) \): \[ f(1) = 2(1)^3 - 3(1)^2 + 7 \] \[ = 2(1) - 3(1) + 7 \] \[ = 2 - 3 + 7 = 6 \]

  3. Calculate \( f(2) \): \[ f(2) = 2(2)^3 - 3(2)^2 + 7 \] \[ = 2(8) - 3(4) + 7 \] \[ = 16 - 12 + 7 = 11 \]

Final results for Question 1:

  • \( f(-1) = 2 \)
  • \( f(1) = 6 \)
  • \( f(2) = 11 \)

Question 2

Given the function \( f(x) = 12x + 32 \):

Now, check the statements given:

  1. \( f(1) = -1 \): \[ f(1) = 12(1) + 32 = 12 + 32 = 44 \quad \text{(False)} \]

  2. \( f(0) = 32 \): \[ f(0) = 12(0) + 32 = 0 + 32 = 32 \quad \text{(True)} \]

  3. \( f(2) = 1 \): \[ f(2) = 12(2) + 32 = 24 + 32 = 56 \quad \text{(False)} \]

  4. \( f(-\frac{1}{2}) = -2 \): \[ f\left(-\frac{1}{2}\right) = 12\left(-\frac{1}{2}\right) + 32 = -6 + 32 = 26 \quad \text{(False)} \]

  5. \( f(4) = 72 \): \[ f(4) = 12(4) + 32 = 48 + 32 = 80 \quad \text{(False)} \]

Final results for Question 2:

  • The only true statement is \( f(0) = 32 \).

Question 3

Given the function \( p(x) = -2(x - 9)^2 + 100 \):

To find the profit when \( x = 15 \): \[ p(15) = -2(15 - 9)^2 + 100 \] \[ = -2(6)^2 + 100 \] \[ = -2(36) + 100 \] \[ = -72 + 100 = 28 \]

Final result for Question 3:

  • The profit when the price of the t-shirts are $15 each is $28.

Summary of Answers

  1. \( f(-1) = 2 \), \( f(1) = 6 \), \( f(2) = 11 \)
  2. The true statement is \( f(0) = 32 \)
  3. Profit from sales if t-shirts are $15 each is $28.