Question

shen is walking. ( d(t) ), given below, is his distance in kilometers from glen city after ( t ) hours of walking. ( d(t) = 11.6 - 4t ) complete the following statements. let ( d^{-1} ) be the inverse function of ( d ). take ( x ) to be an output of the function ( d ). that is, ( x = d(t) ) and ( t = d^{-1}(x) ). (a) which statement best describes ( d^{-1}(x) )?

• the reciprocal of his distance from glen city (in kilometers) after walking ( x ) hours.
• the amount of time he has walked (in hours) when he is ( x ) kilometers from glen city.
• his distance from glen city (in kilometers) after he has walked ( x ) hours.
• the ratio of the amount of time he has walked (in hours) to his distance from glen city (in kilometers), ( x ).

(b) ( d^{-1}(x) = )
(c) ( d^{-1}(6.4) = )

Explanation:

Part (a)

Step1: Understand \( D(t) \)

\( D(t) = 11.6 - 4t \) represents distance from Glen City after \( t \) hours. So \( D(t) \) (let's call it \( x \)) is distance, and \( t \) is time. The inverse function \( D^{-1}(x) \) should take distance \( x \) and give time \( t \).

Step2: Analyze Options

• Option 1: "reciprocal" is wrong (inverse function, not reciprocal of \( D(t) \)).
• Option 2: "time when distance is \( x \)" – since \( D(t) = x \) means \( 11.6 - 4t = x \), solving for \( t \) gives time for distance \( x \). This matches.
• Option 3: "distance after \( x \) hours" is \( D(x) \), not inverse.
• Option 4: "ratio of time to distance" is not what inverse function does.

Step1: Start with \( x = D(t) \)

Given \( D(t) = 11.6 - 4t \), so \( x = 11.6 - 4t \).

Step2: Solve for \( t \) (which is \( D^{-1}(x) \))

Subtract \( 11.6 \): \( x - 11.6 = -4t \)
Multiply by \( -1 \): \( 11.6 - x = 4t \)
Divide by 4: \( t = \frac{11.6 - x}{4} = \frac{11.6}{4} - \frac{x}{4} = 2.9 - 0.25x \)
So \( D^{-1}(x) = \frac{11.6 - x}{4} \) (or simplified as \( 2.9 - 0.25x \))

Step1: Substitute \( x = 6.4 \) into \( D^{-1}(x) \)

From part (b), \( D^{-1}(x) = \frac{11.6 - x}{4} \). Substitute \( x = 6.4 \):

Step2: Calculate

\( D^{-1}(6.4) = \frac{11.6 - 6.4}{4} = \frac{5.2}{4} = 1.3 \)

Answer:

The amount of time he has walked (in hours) when he is \( x \) kilometers from Glen City. (The second option in the list for part (a))

Part (b)