Final answer:
Elevation function: f(x)=(x−1)^2 +50, positive x values represent distance hiked north with f(3)=54 feet, and negative x values represent distance hiked south with f(−2)=59 feet, overall elevation increases; domain is x≥0.
Step-by-step explanation:
The function f(x)=(x−1)^2 +50 represents Melanie's elevation, in feet, based on the distance she has hiked north (positive x values).
To find the elevation if Melanie hiked south (negative x values), you would substitute negative x values into the function.
Let's consider a specific example:
if Melanie hiked 3 miles north ( x=3), her elevation would be:
f(3)=(3−1)^2 +50
=2^2 +50
=4+50
=54
So, if Melanie hiked 3 miles north, her elevation would be 54 feet.
Now, to find the elevation if Melanie hiked south, you could use a negative x value.
For example, if Melanie hiked 2 miles south (x=−2):
f(−2)=((−2)−1)^2 +50
=(−3)^2 +50
=9+50
=59
So, if Melanie hiked 2 miles south, her elevation would be 59 feet.
Interpretation: The function f(x)=(x−1)^2 +50 represents a parabolic relationship between the distance Melanie has hiked north and her elevation.
The squared term (x−1)^2 ensures that the function is always positive, and the constant term 50 adds a base elevation. As Melanie hikes north, her elevation increases, and as she hikes south, her elevation also increases, but at a different rate.
The domain of the function is the set of all possible input values (in this case, distances Melanie has hiked north). Since Melanie can hike any positive distance north, the domain is all real numbers where x is greater than or equal to zero. Therefore, the appropriate domain for the function is x≥0.