Question

Given the function f(x) = 0.5(3)x, what is the value of f−1(7)? (2 points)

Answer 1

To find the value of f^(-1)(7), we need to determine the inverse function of f(x) = 0.5(3)^x.

To find the inverse function, we interchange the roles of x and y and solve for y.

Let's start with the original function:

y = 0.5(3)^x

Now, let's interchange x and y:

x = 0.5(3)^y

Next, solve for y:

x = 0.5(3)^y

2x = 3^y

log base 3 (2x) = y

So, the inverse function of f(x) = 0.5(3)^x is:

f^(-1)(x) = log base 3 (2x)

Now, we can find the value of f^(-1)(7):

f^(-1)(7) = log base 3 (2 * 7)

= log base 3 (14)

Using a calculator, we can approximate the value of log base 3 (14) to be approximately 2.264.

Therefore, the value of f^(-1)(7) is approximately 2.264.