Guadalupe is on a hiking trail that goes north to south. If Guadalupe hikes $$ x $$ miles north, her elevation, in feet, can be found using the function $$ f(x)=(x+2)^{2}+150 $$. Negative $$ x $$ values would find the elevation if Guadalupe hiked south. Find and interpret the given function values and determine an appropriate domain for the function. Answer Altemph out or ? $$ f(-3)=\square $$, means if Guadalupe hikes 3 miles to the feet. This interpretation $$ f(6)=\square $$, means if Guadalupe hikes 6 miles to the This interpretation $$ f(7,5)=\square $$, her elevation would be $$ \square $$ feet. This interpretation Based on the observations above, it is clear that an appropriate domain for the function is of the problem,

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Below is a step‐by‐step solution.

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### 1. Evaluate $$ f(-3) $$

The function is given by

$$
f(x)=(x+2)^2+150.
$$

Substitute $$ x=-3 $$:

$$
f(-3)=((-3)+2)^2+150.
$$

Calculate inside the parentheses:

$$
-3+2=-1.
$$

Then square the result:

$$
(-1)^2=1.
$$

Finally, add 150:

$$
1+150=151.
$$

**Interpretation:**  
$$ f(-3)=151 $$ means if Guadalupe hikes 3 miles to the **south** (since negative $$ x $$ indicates a southern direction), her elevation will be **151 feet**.

---

### 2. Evaluate $$ f(6) $$

Substitute $$ x=6 $$:

$$
f(6)=(6+2)^2+150.
$$

Simplify inside the parentheses:

$$
6+2=8.
$$

Then square 8:

$$
8^2=64.
$$

Finally, add 150:

$$
64+150=214.
$$

**Interpretation:**  
$$ f(6)=214 $$ means if Guadalupe hikes 6 miles to the **north** (since positive $$ x $$ denotes north), her elevation will be **214 feet**.

---

### 3. Evaluate $$ f(7.5) $$

Substitute $$ x=7.5 $$:

$$
f(7.5)=(7.5+2)^2+150.
$$

Simplify the expression inside the parentheses:

$$
7.5+2=9.5.
$$

Then square 9.5:

$$
(9.5)^2=90.25.
$$

Finally, add 150:

$$
90.25+150=240.25.
$$

**Interpretation:**  
$$ f(7.5)=240.25 $$ means if Guadalupe hikes 7.5 miles to the **north**, her elevation will be **240.25 feet**.

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### 4. Determine an Appropriate Domain

In the context of the problem, the variable $$ x $$ represents the number of miles Guadalupe hikes from a fixed starting point, where:
- A positive $$ x $$ indicates a hike to the **north**.
- A negative $$ x $$ indicates a hike to the **south**.

Since there is no additional information restricting how far she can go on the trail (other than the finite length of any real trail), a general interpretation is to allow any realistic displacement. Thus, mathematically, we can take the **domain** as

$$
(-\infty, \infty)
$$

which represents all real numbers. However, if the trail had known endpoints, the domain would be limited to the interval corresponding to the actual trail length. Without further restrictions, we choose

$$
\text{Domain: } x \in (-\infty, \infty).
$$

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### Final Answers

- $$ f(-3)=151 $$: If Guadalupe hikes 3 miles **south**, her elevation is **151 feet**.
- $$ f(6)=214 $$: If Guadalupe hikes 6 miles **north**, her elevation is **214 feet**.
- $$ f(7.5)=240.25 $$: If Guadalupe hikes 7.5 miles **north**, her elevation is **240.25 feet**.
- An appropriate domain for the function is $$ x \in (-\infty, \infty) $$ (or more specifically, the set of distances where she is actually able to hike on the trail).
$$ f(-3)=151 $$ means if Guadalupe hikes 3 miles south, her elevation is 151 feet. $$ f(6)=214 $$ means if she hikes 6 miles north, her elevation is 214 feet. $$ f(7.5)=240.25 $$ means hiking 7.5 miles north results in an elevation of 240.25 feet. An appropriate domain for the function is all real numbers, representing any distance she can hike on the trail.

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