Final answer:
The inverse of f(x) = 6*5^(x+1/4) - 3 can be found by swapping the x and y variables and solving for y step by step. The inverse function is f^-1(x) = log_5((x + 3)/6) - 1/4.
Step-by-step explanation:
The inverse of a function can be found by swapping the x and y variables and solving for y. To find the inverse of f(x) = 6*5^(x+1/4) - 3, we start by replacing f(x) with y. Then, swap the x and y variables:
x = 6*5^(y+1/4) - 3
Next, solve for y. Start by adding 3 to both sides:
x + 3 = 6*5^(y+1/4)
Then, divide both sides by 6:
(x + 3)/6 = 5^(y+1/4)
To isolate the exponent, take the log base 5 of both sides:
log5((x + 3)/6) = y + 1/4
Subtract 1/4 from both sides:
log5((x + 3)/6) - 1/4 = y
Finally, swap y back with f-1(x):
f-1(x) = log5((x + 3)/6) - 1/4