Final answer:
The inverse of the function f(x) = (3x-24)^4 is f⁻¹(y) = (√[4]{y} + 24) / 3. It is a function with a domain of all real numbers ≥ 0 and a range that includes all non-negative numbers.
Step-by-step explanation:
To find the inverse of the function f(x) = (3x-24)^4, you first replace f(x) with y:
y = (3x-24)^4
Now to find the inverse, you need to solve for x:
Take the fourth root of both sides:
√[4]{y} = 3x - 24
Now, isolate x:
3x = √[4]{y} + 24
x = (√[4]{y} + 24) / 3
The inverse function, then, is f⁻¹(y) = (√[4]{y} + 24) / 3. Now, we determine if this is a function. Since for every value of y, there is exactly one corresponding value of x, f⁻¹(y) is a function.
The domain of the inverse function is all real numbers ≥ 0, as only non-negative numbers can be the fourth root, making the range of the original function also all non-negative numbers.