Question

If f, of, xf(x) is an exponential function where f, of, minus, 1, equals, 30f(−1)=30 and f, of, 4, equals, 85f(4)=85, then find the value of f, of, 3f(3), to the nearest hundredth.

Answer 1

To find the value of f(3f(3)), we first need to find the value of f(3).

Given that f(x) is an exponential function, we can rewrite the given information as:

f(-1) = 30f(0) = 30
f(4) = 85f(0) = 85

From the first equation, we can solve for f(0):

30f(0) = 30
f(0) = 1

Now we have the values of f(-1) = 30 and f(0) = 1. We can use these to find the base of the exponential function.

Using f(x) = ab^x, we can set up two equations using the values we have:

30 = ab^(-1)
85 = ab^4

Dividing the two equations, we get:

(30/85) = (ab^(-1))/(ab^4)
(30/85) = (b^(-1))/(b^4)

Simplifying the right side, we get:

(30/85) = b^(-1-4)
(30/85) = b^(-5)

Take the reciprocal of both sides to solve for b:

(85/30) = b^5
17/6 = b^5

Taking the fifth root of both sides, we get:

b = (17/6)^(1/5)

Now that we have found the base b, we can find f(3) by plugging x = 3 into the equation f(x) = ab^x:

f(3) = (1)(17/6)^(1/5)^3
f(3) = (17/6)^(3/5) ≈ 2.657

Finally, we can find f(3f(3)) by plugging f(3) into the equation f(x) = ab^x:

f(3f(3)) = (17/6)^(2.657)
f(3f(3)) ≈ 12.54

Therefore, the value of f(3f(3)) to the nearest hundredth is 12.54.