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Over [174]
3 years ago
5

Graph y = -­4x2 ­- 2 and its inverse.

Mathematics
1 answer:
CaHeK987 [17]3 years ago
5 0

Be careful with finding its inverse! An inverse only exists for one-to-one functions so we need to restrict the domain such that the quadratic is one-to-one. For brevity, we'll take the domain: x \geq 0 but you could just as easily take x \leq 0.

Being that the quadratic is now one-to-one, we'll try to find its inverse.

By definition, an inverse function is the function flipped across the line y = x. So being that, we need to interchange the x and y coordinates around and then solve for y.

This gives us:

x = -4y^2 - 2 \\ x + 2 = -4y^2 \\ -\frac{1}{4}\left(x + 2\right) = y^2 \\ y = \sqrt{-\frac{1}{4}\left(x + 2\right)}

Now, this function is defined for when y \geq 0 and x \leq -2.

So I'll end with this, what is the relationship between the domain/range of the original function and the domain/range of the inverse?

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The formula below, Newton's law of cooling, allows us to calculate the temperature of an object as it cools. T = R + (T0 − R)e-k
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The Newton`s law of cooling:
T = R + ( To - R ) * e ^(-kt )
We know that: T = 32° F ( the water freezes at 32° F ), To = 55° F, R = 5° F
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33° F = 5° F + ( 55° F - 5° F ) · 2.72^(-0.012 t )
33 - 5 = 50 · 2.72^(-0.012 t)
27 = 50 · 2.72^(-0.012 t)
2.72^(-0.012 t) = 27/50
1 / 2.72^(0.012 t) = 0.54
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0.012 t = log(base e) 1.85 = ln 1.85 = 0.6
t = 0.6 : 0.012 = 50
Answer: It would take 50 minutes.
 

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