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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Let’s break that down with a sprinkle of fun! To find \( f(-3) \): Plugging in -3 gives us \( f(-3) = ((-3) + 2)^2 + 150 = (-1)^2 + 150 = 1 + 150 = 151 \). So, if Guadalupe hikes 3 miles south, her elevation would be 151 feet. This means she's still got a pretty nice view! Now, for \( f(6) \): Plugging in 6 results in \( f(6) = (6 + 2)^2 + 150 = (8)^2 + 150 = 64 + 150 = 214 \). That means if she hikes 6 miles north, she’ll be at an elevation of 214 feet, climbing higher than a bear in a tree! Finally, when we evaluate \( f(7.5) \): We get \( f(7.5) = (7.5 + 2)^2 + 150 = (9.5)^2 + 150 = 90.25 + 150 = 240.25 \). So if she hiked 7.5 miles north, her elevation would be \( 240.25 \) feet, soaring like an eagle! As for the domain of this function, since negative values represent hiking south and positive values represent hiking north, we can say the appropriate domain is all real numbers, reflecting that Guadalupe can hike both directions on the trail! What an adventure!
