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[tex]$L(t) = 1.25 t + 82$[/tex]

Complete the following statements.

Let [tex]$L^{-1}$[/tex] be the inverse function of [tex]$L$[/tex]. Take [tex]$x$[/tex] to be an output of the function [tex]$L$[/tex]. That is, [tex]$x = L(t)$[/tex] and [tex]$t = L^{-1}(x)$[/tex].

(a) Which statement best describes [tex]$L^{-1}(x)$[/tex]?
1. The amount of time (in minutes) it takes to have [tex]$x$[/tex] liters of liquid.
2. The amount of liquid (in liters) after [tex]$x$[/tex] minutes.
3. The ratio of the amount of liquid (in liters) to the number of minutes, [tex]$x$[/tex].
4. The reciprocal of the amount of liquid (in liters) after [tex]$x$[/tex] minutes.

(b) [tex]$L^{-1}(x) = \square$[/tex]

(c) [tex]$L^{-1}(130) = \square$[/tex]

Sagot :

GinnyAnsweridn
To address the given problem, let's break it down step by step.

We're given the function [tex]\( L(t) = 1.25t + 82 \)[/tex]. We need to find the inverse function [tex]\( L^{-1}(x) \)[/tex] and then use that inverse function to complete parts (a), (b), and (c) of the problem.

### Part (a)
Description of [tex]\( L^{-1}(x) \)[/tex]:

To determine the description that best fits [tex]\( L^{-1}(x) \)[/tex], it is essential to understand what an inverse function represents. Given [tex]\( x = L(t) \)[/tex], which translates to [tex]\( x \)[/tex] being the amount of liquid after [tex]\( t \)[/tex] minutes, the inverse [tex]\( L^{-1}(x) \)[/tex] yields the value of [tex]\( t \)[/tex] for a given amount of liquid [tex]\( x \)[/tex].

In other words, [tex]\( L^{-1}(x) \)[/tex] provides the time it takes to obtain [tex]\( x \)[/tex] liters of liquid.

Thus, the correct statement is:
- The amount of time (in minutes) it takes to have [tex]\( x \)[/tex] liters of liquid.

### Part (b)
Finding [tex]\( L^{-1}(x) \)[/tex]:

To find the inverse function [tex]\( L^{-1}(x) \)[/tex], we start with the original function:
[tex]\[ L(t) = 1.25t + 82 \][/tex]

We need to solve for [tex]\( t \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ x = 1.25t + 82 \][/tex]

Subtract 82 from both sides of the equation:
[tex]\[ x - 82 = 1.25t \][/tex]

Next, divide both sides by 1.25 to isolate [tex]\( t \)[/tex]:
[tex]\[ t = \frac{x - 82}{1.25} \][/tex]

So, the inverse function [tex]\( L^{-1}(x) \)[/tex] is:
[tex]\[ L^{-1}(x) = \frac{x - 82}{1.25} \][/tex]

Simplifying this expression, we get:
[tex]\[ L^{-1}(x) = 0.8x - 65.6 \][/tex]

### Part (c)
Calculating [tex]\( L^{-1}(130) \)[/tex]:

To find [tex]\( L^{-1}(130) \)[/tex], we substitute [tex]\( x = 130 \)[/tex] into the inverse function [tex]\( L^{-1}(x) \)[/tex]:
[tex]\[ L^{-1}(130) = 0.8 \cdot 130 - 65.6 \][/tex]

First, calculate [tex]\( 0.8 \cdot 130 \)[/tex]:
[tex]\[ 0.8 \cdot 130 = 104 \][/tex]

Then, subtract 65.6:
[tex]\[ 104 - 65.6 = 38.4 \][/tex]

Thus:
[tex]\[ L^{-1}(130) = 38.4 \][/tex]

### Summary
Here are the answers to the questions:

(a) The amount of time (in minutes) it takes to have [tex]\( x \)[/tex] liters of liquid. (Choice 1)

(b) [tex]\( L^{-1}(x) = 0.8x - 65.6 \)[/tex]

(c) [tex]\( L^{-1}(130) = 38.4 \)[/tex]
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