To find the inverse of the function f(x) = 6.5^x + 1/4 - 3, one must isolate the exponential term and then use the natural logarithm to solve for x, resulting in the inverse function.
Step-by-step explanation:
To find the inverse of the function f(x) = 6.5^x + 1/4 - 3, we need to solve for x in terms of y, where y = f(x). The original function involves exponentiation, addition, and subtraction. The critical step in finding the inverse involves taking the natural logarithm to isolate the exponential term. Following this approach, let's go through the steps:
First, replace f(x) with y: y = 6.5^x + 1/4 - 3.
Then, add 3 to both sides to isolate the exponential term: y + 3 = 6.5^x + 1/4.
Next, subtract 1/4 from both sides: y + 3 - 1/4 = 6.5^x.
Now, take the natural logarithm of both sides to cancel the exponential function: ln(y + 3 - 1/4) = ln(6.5^x). This becomes ln(y + 3 - 1/4) = x ln(6.5).
Finally, solve for x by dividing both sides by ln(6.5): x = ln(y + 3 - 1/4) / ln(6.5).
This final expression gives us the inverse function of f(x).