Keep in mind: This is Calculus, not Pre-Calculus. This involves finding the inverse of a function, completing the square, and analyzing the domain of a function. In this problem, we are given a function P(x) that represents the price of a media player as a function of the weekly sales x. To find the inverse function P-¹(x), we need to solve for x in terms of P(x). This requires completing the square, which is a technique from calculus used to transform a quadratic expression into a perfect square. Completing the square involves dividing the coefficient of the x-term by 2, squaring the result, and adding and subtracting this value inside the parentheses. This technique is used to simplify the expression and make it easier to solve for x. Once we have found the inverse function P-¹(x), we need to analyze its domain, which is the set of all possible values of x that make the expression inside the square root non-negative. This involves solving an inequality that typically involves algebraic manipulation and the use of calculus techniques, such as solving inequalities and simplifying expressions. Finally, the problem also involves the composition of functions, which is a fundamental concept in calculus. Composing functions involves applying one function to the output of another function. In this problem, we need to compose the function h(t) = -2t + 1 with its inverse function h-¹(t) = (1 - t)/2 to obtain the identity function. This process requires substituting one function into the other and simplifying the resulting expression.
Answer:
a. P-¹(x) = 11.67 ± sqrt((9500 - x)/15), and the domain of P-¹(x) is (-∞, 9500].
b. P-¹(105) = 41.67, which represents the weekly sales required to achieve a price of $105 per media player.
c. P-¹(P(x)) = 11.67 ± sqrt((9500 - P(x))/15), where P(x) must be in the domain of P-¹(x), which is (-∞, 9500].
d. The inverse of h(t) = -2t + 1 is h-¹(t) = (1 - t)/2. Composing h(t) and h-¹(t) gives the identity function, h(h-¹(t)) = t.
Explanation:
a. To find the inverse of the function P(x), we need to solve for x in terms of P(x). The equation is:
P(x) = -15x² + 350x - 2000
Let y = P(x), then we have:
y = -15x² + 350x - 2000
To solve for x, we need to complete the square:
y = -15(x² - 23.33x + 766.67) - 2000 + 11500
y = -15(x - 11.67)² + 9500
Dividing both sides by -15, we get:
(x - 11.67)² = (9500 - y)/15
x - 11.67 = ±sqrt((9500 - y)/15)
x = 11.67 ± sqrt((9500 - y)/15)
So, the inverse function of P(x) is:
P-¹(x) = 11.67 ± sqrt((9500 - x)/15)
The domain of P-¹(x) is the set of all values of x that make the expression inside the square root non-negative, i.e., (9500 - x)/15 ≥ 0. Solving for x, we get x ≤ 9500.
Therefore, the domain of P-¹(x) is (-∞, 9500].
b. To find P-¹(105), we substitute y = 105 into the formula for P-¹(x):
P-¹(105) = 11.67 ± sqrt((9500 - 105)/15)
P-¹(105) = 11.67 ± 30
So, P-¹(105) can be either 41.67 or -8.33.
Interpretation: P-¹(105) represents the weekly sales required to achieve a price of $105 per media player. According to the inverse function, there are two possible values: 41.67 and -8.33. However, since the domain of P-¹(x) is restricted to (-∞, 9500], only the value 41.67 is valid.
c. To find P-¹(P(x)), we substitute P(x) for x in the formula for P-¹(x):
P-¹(P(x)) = 11.67 ± sqrt((9500 - P(x))/15)
Note that P(x) must be in the domain of P-¹(x), which is (-∞, 9500].
d. To find the inverse of h(t) = -2t + 1, we need to solve for t in terms of h(t):
h(t) = -2t + 1
y = -2t + 1
-2t = y - 1
t = (1 - y)/2
So, the inverse function of h(t) is:
h-¹(t) = (1 - t)/2
To compose h(t) and h-¹(t), we substitute h-¹(t) for t in the formula for h(t):
h(h-¹(t)) = -2h-¹(t) + 1
h(h-¹(t)) = -2(1 - t)/2 + 1
h(h-¹(t)) = -1 + t + 1
h(h-¹(t)) = t
Therefore, the composition of h(t) and h-¹(t) is the identity function, h(h-¹(t)) = t.