High School

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A function \( f \) has the form \( f(x) = ae^{kx} \). Find \( f \) if it is known that \( f(0) = 100 \) and \( f(1) = 170 \).

(Hint: \( e^{kx} = (e^k)^x \))

Answer :

Final answer:

The function f has the form f(x) = 100(e^x).

Explanation:

Given that the function f has the exponential form f(x) = aekx, we are provided with two points on the graph of this function, namely f(0) = 100 and f(1) = 170. Using these values, we can find the constants a and k.

First, let's find k. To do this, we can use the fact that ekx = (ek)x. Setting x = 1 in this equation and using the second given point, we have:

170 = ae^1

Dividing both sides by a, we get:

170/a = e^1

Taking natural logarithms of both sides, we have:

ln(170/a) = ln(e^1)

Using the fact that ln(e^x) = x, we have:

ln(170/a) = 1

Taking exponentials of both sides, we have:

k = e^1 = 2.71828... (rounded to 4 decimal places)

Next, let's find a. Using the first given point f(0) = 100, we have:

100 = ae^0

Simplifying, we have:

a = 100/e^0 = 100 (since e^0 is equal to 1)

Putting it all together, our function is:

f(x) = aekx = 100e^x = 100(e^x)

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