Math / Algebra

QuestionSalma is walking. $D(t)$ , given below, is her distance in kilometers from Glen City after $t$ hours of walking. $D(t)=12.8-4 t$
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 ratio of the amount of time she has walked (in hours) to her distance from Glen City (in kilometers), $x$ .
The reciprocal of her distance from Glen City (in kilometers) after walking $x$ hours.
The amount of time she has walked (in hours) when she is $x$ kilometers from Glen City. Her distance from Glen City (in kilometers) after she has walked $x$ hours. (b) $D^{-1}(x)=$ $\square$ (c) $D^{-1}(8.4)=$ $\square$

Studdy Solution

STEP 1

What is this asking? Given Salma's distance from Glen City over time, how much time does it take her to reach a certain distance? Watch out! Don't mix up the input and output of the inverse function! $D(t)$ tells us the distance after a given time, but $D^{-1}(x)$ tells us the time it takes to reach a given distance.

STEP 2

1. Understand the Function
2. Find the Inverse Function
3. Calculate the Specific Time

STEP 3

Alright, so we've got this function $D(t) = 12.8 - 4t$ , where $t$ is the time Salma's been walking in hours, and $D(t)$ is her distance from Glen City in kilometers.
So, if we plug in a time, it tells us how far she is from Glen City at that time!

STEP 4

For example, when $t = 0$ (when she starts walking), her distance is $D(0) = 12.8 - 4 \cdot 0 = 12.8$ kilometers.
This means Salma starts 12.8 kilometers away from Glen City.

STEP 5

The $-4t$ part tells us that for every hour ( $t=1$ ) she walks, her distance from Glen City decreases by 4 kilometers.
That's our rate of change!

STEP 6

Now, we want to find the inverse function, $D^{-1}(x)$ .
This function will do the opposite of $D(t)$ .
We'll give it a distance, $x$ , and it will tell us the time it took Salma to get that far from Glen City.

STEP 7

To find the inverse, we swap $t$ and $x$ in our original equation: $x = 12.8 - 4t$ .
Now, we solve for $t$ .

STEP 8

First, we want to isolate the term with $t$ .
We can do this by subtracting 12.8 from both sides of the equation: $x - 12.8 = -4t$ .

STEP 9

Next, we divide both sides by $-4$ to get $t$ by itself: $\frac{x - 12.8}{-4} = t$ .

STEP 10

So, our inverse function is $D^{-1}(x) = \frac{x - 12.8}{-4}$ .
This tells us the time it takes Salma to be $x$ kilometers from Glen City.

STEP 11

We want to find $D^{-1}(8.4)$ , which means we want to know how long it takes Salma to be 8.4 kilometers from Glen City.

STEP 12

We plug in $x = 8.4$ into our inverse function: $D^{-1}(8.4) = \frac{8.4 - 12.8}{-4}$ .

STEP 13

Now we calculate: $D^{-1}(8.4) = \frac{-4.4}{-4} = 1.1$ .

STEP 14

So, it takes Salma 1.1 hours to be 8.4 kilometers from Glen City.

STEP 15

(a) The amount of time she has walked (in hours) when she is $x$ kilometers from Glen City. (b) $D^{-1}(x) = \frac{x - 12.8}{-4}$ (c) $D^{-1}(8.4) = 1.1$

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Studdy Solution

STEP 1

STEP 2

1. Understand the Function
2. Find the Inverse Function
3. Calculate the Specific Time

STEP 3

Alright, so we've got this function $D(t) = 12.8 - 4t$ , where $t$ is the time Salma's been walking in hours, and $D(t)$ is her distance from Glen City in kilometers.
So, if we plug in a time, it tells us how far she is from Glen City at that time!

STEP 4

For example, when $t = 0$ (when she starts walking), her distance is $D(0) = 12.8 - 4 \cdot 0 = 12.8$ kilometers.
This means Salma starts 12.8 kilometers away from Glen City.

STEP 5

The $-4t$ part tells us that for every hour ( $t=1$ ) she walks, her distance from Glen City decreases by 4 kilometers.
That's our rate of change!

STEP 6

Now, we want to find the inverse function, $D^{-1}(x)$ .
This function will do the opposite of $D(t)$ .
We'll give it a distance, $x$ , and it will tell us the time it took Salma to get that far from Glen City.

STEP 7

To find the inverse, we swap $t$ and $x$ in our original equation: $x = 12.8 - 4t$ .
Now, we solve for $t$ .

STEP 8

First, we want to isolate the term with $t$ .
We can do this by subtracting 12.8 from both sides of the equation: $x - 12.8 = -4t$ .

STEP 9

Next, we divide both sides by $-4$ to get $t$ by itself: $\frac{x - 12.8}{-4} = t$ .

STEP 10

So, our inverse function is $D^{-1}(x) = \frac{x - 12.8}{-4}$ .
This tells us the time it takes Salma to be $x$ kilometers from Glen City.

STEP 11

We want to find $D^{-1}(8.4)$ , which means we want to know how long it takes Salma to be 8.4 kilometers from Glen City.

STEP 12

We plug in $x = 8.4$ into our inverse function: $D^{-1}(8.4) = \frac{8.4 - 12.8}{-4}$ .

STEP 13

Now we calculate: $D^{-1}(8.4) = \frac{-4.4}{-4} = 1.1$ .

STEP 14

So, it takes Salma 1.1 hours to be 8.4 kilometers from Glen City.

STEP 15

(a) The amount of time she has walked (in hours) when she is $x$ kilometers from Glen City. (b) $D^{-1}(x) = \frac{x - 12.8}{-4}$ (c) $D^{-1}(8.4) = 1.1$

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Studdy Solution

STEP 1

STEP 2

1. Understand the Function2. Find the Inverse Function3. Calculate the Specific Time

STEP 3

Alright, so we've got this function D(t)=12.8−4tD(t) = 12.8 - 4tD(t)=12.8−4t, where ttt is the **time** Salma's been walking in hours, and D(t)D(t)D(t) is her **distance** from Glen City in kilometers.So, if we plug in a time, it tells us how far she is from Glen City at that time!

STEP 4

For example, when t=0t = 0t=0 (when she starts walking), her distance is D(0)=12.8−4⋅0=12.8D(0) = 12.8 - 4 \cdot 0 = 12.8D(0)=12.8−4⋅0=12.8 kilometers.This means Salma **starts** 12.8 kilometers away from Glen City.

STEP 5

The −4t-4t−4t part tells us that for every hour (t=1t=1t=1) she walks, her distance from Glen City *decreases* by 4 kilometers.That's our **rate of change**!

STEP 6

Now, we want to find the *inverse* function, D−1(x)D^{-1}(x)D−1(x).This function will do the *opposite* of D(t)D(t)D(t).We'll give it a **distance**, xxx, and it will tell us the **time** it took Salma to get that far from Glen City.

STEP 7

To find the inverse, we **swap** ttt and xxx in our original equation: x=12.8−4tx = 12.8 - 4tx=12.8−4t.Now, we **solve for** ttt.

STEP 8

First, we want to **isolate** the term with ttt.We can do this by **subtracting** 12.8 from both sides of the equation: x−12.8=−4tx - 12.8 = -4tx−12.8=−4t.

STEP 9

Next, we **divide both sides** by −4-4−4 to get ttt by itself: x−12.8−4=t\frac{x - 12.8}{-4} = t−4x−12.8​=t.

STEP 10

So, our inverse function is D−1(x)=x−12.8−4D^{-1}(x) = \frac{x - 12.8}{-4}D−1(x)=−4x−12.8​.This tells us the **time** it takes Salma to be xxx kilometers from Glen City.

STEP 11

We want to find D−1(8.4)D^{-1}(8.4)D−1(8.4), which means we want to know how long it takes Salma to be **8.4 kilometers** from Glen City.

STEP 12

We **plug in** x=8.4x = 8.4x=8.4 into our inverse function: D−1(8.4)=8.4−12.8−4D^{-1}(8.4) = \frac{8.4 - 12.8}{-4}D−1(8.4)=−48.4−12.8​.

STEP 13

Now we **calculate**: D−1(8.4)=−4.4−4=1.1D^{-1}(8.4) = \frac{-4.4}{-4} = 1.1D−1(8.4)=−4−4.4​=1.1.

STEP 14

So, it takes Salma **1.1 hours** to be 8.4 kilometers from Glen City.

STEP 15

(a) The amount of time she has walked (in hours) when she is xxx kilometers from Glen City. (b) D−1(x)=x−12.8−4D^{-1}(x) = \frac{x - 12.8}{-4}D−1(x)=−4x−12.8​ (c) D−1(8.4)=1.1D^{-1}(8.4) = 1.1D−1(8.4)=1.1

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1. Understand the Function
2. Find the Inverse Function
3. Calculate the Specific Time

Alright, so we've got this function $D(t) = 12.8 - 4t$ , where $t$ is the time Salma's been walking in hours, and $D(t)$ is her distance from Glen City in kilometers.
So, if we plug in a time, it tells us how far she is from Glen City at that time!

For example, when $t = 0$ (when she starts walking), her distance is $D(0) = 12.8 - 4 \cdot 0 = 12.8$ kilometers.
This means Salma starts 12.8 kilometers away from Glen City.

The $-4t$ part tells us that for every hour ( $t=1$ ) she walks, her distance from Glen City decreases by 4 kilometers.
That's our rate of change!

Now, we want to find the inverse function, $D^{-1}(x)$ .
This function will do the opposite of $D(t)$ .
We'll give it a distance, $x$ , and it will tell us the time it took Salma to get that far from Glen City.

To find the inverse, we swap $t$ and $x$ in our original equation: $x = 12.8 - 4t$ .
Now, we solve for $t$ .

First, we want to isolate the term with $t$ .
We can do this by subtracting 12.8 from both sides of the equation: $x - 12.8 = -4t$ .

Next, we divide both sides by $-4$ to get $t$ by itself: $\frac{x - 12.8}{-4} = t$ .

So, our inverse function is $D^{-1}(x) = \frac{x - 12.8}{-4}$ .
This tells us the time it takes Salma to be $x$ kilometers from Glen City.

We want to find $D^{-1}(8.4)$ , which means we want to know how long it takes Salma to be 8.4 kilometers from Glen City.

We plug in $x = 8.4$ into our inverse function: $D^{-1}(8.4) = \frac{8.4 - 12.8}{-4}$ .

Now we calculate: $D^{-1}(8.4) = \frac{-4.4}{-4} = 1.1$ .

So, it takes Salma 1.1 hours to be 8.4 kilometers from Glen City.

(a) The amount of time she has walked (in hours) when she is $x$ kilometers from Glen City. (b) $D^{-1}(x) = \frac{x - 12.8}{-4}$ (c) $D^{-1}(8.4) = 1.1$