Answer :
Final answer:
The domain of f(x) = 4+ log5 (6-7x) is (-∞, 6/7). The inverse of f(x), denoted as f¹(x), is (6 - 5^(x-4))/7.
Explanation:
To solve this question, we need to apply the principles of logarithms and functions.
a.) The domain of f(x): The domain of a function is the set of all possible input values (x-values). For the logarithm function, the argument (the part inside the log) must be positive. So, the inequality that we need to solve to find the domain is 6-7x > 0.
Solving for x gives: x < 6/7.
So, the domain of f(x) in interval notation is (-∞, 6/7).
b.) Find the inverse f¹(x): To find the inverse of a function, we interchange the x and y (or f(x)) in the function and then solve for y.
So, if y = 4 + log5 (6-7x), the inverse function is found by swapping x and y to get: x = 4 + log5 (6-7y).
We can then solve for y to get the inverse in simplified form:
5^(x-4) = 6 - 7y, y = (6 - 5^(x-4))/7.
So, f¹(x) = (6 - 5^(x-4))/7.
Learn more about Logarithms and functions here:
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