If f, of, xf(x) is an exponential function where f, of, minus, 2, equals, 15f(−2)=15 and f, of, 5, point, 5, equals, 1f(5.5)=1, then find the value of f, of, 7f(7), to the nearest hundredth.
To find the value of f(7), we need to use the given information about f(-2) and f(5.5) to determine the specific exponential function.
First, let's use the information about f(−2). We know that f(−2) = 15.
Using the general form of an exponential function, y = a * b^x, where a is the initial value and b is the base, we can substitute the given values to form an equation:
15 = a * b^(-2)
Next, let's use the information about f(5.5). We know that f(5.5) = 1.
Substituting the values into the general form equation:
1 = a * b^5.5
Now we have a system of two equations:
15 = a * b^(-2)
1 = a * b^5.5
To solve this system, we can divide the second equation by the first equation:
1/15 = b^(5.5-(-2))
1/15 = b^7.5
Taking the 7.5th root of both sides to isolate b:
(b^7.5)^(1/7.5) = (1/15)^(1/7.5)
b = (1/15)^(1/7.5)
Now that we have the value of b, we can substitute it back into either of the original equations to solve for a. Let's use the first equation:
15 = a * b^(-2)
15 = a * [(1/15)^(1/7.5)]^(-2)
15 = a * 15^(-2/7.5)
Simplifying:
15 = a * 15^(-2/7.5)
a = 15 / 15^(-2/7.5)
a = 15^(1 + 2/7.5)
a ≈ 15^(1.267)
Now we have the values of a and b, so we can form the exponential function:
f(x) = a * b^x
f(x) = 15^(1.267) * [(1/15)^(1/7.5)]^x
To find the value of f(7), we substitute x = 7 into the function:
f(7) ≈ 15^(1.267) * [(1/15)^(1/7.5)]^7
Using a calculator to evaluate this expression:
f(7) ≈ 0.091
So, the value of f(7) to the nearest hundredth is 0.09.