f⁻¹(x)= (x-4)³+1
f(x)= 4+³√(x-1)
To find the inverse of this function, we need to switch the roles of x and f(x) and solve for x.
Let y = f(x) = (x – 4)³ + 1
Then, we can solve for x:
y – 1 = (x – 4)³
∛(y – 1) = x – 4
x = ∛(y – 1) + 4
So, the inverse function is:
f⁻¹(x) = ∛(x – 1) + 4
Now we can verify that the composition of the functions f and f⁻¹ (or f⁻¹ and f) give the input/output pairs (x, x) for all x in the domain of each function.
f(f⁻¹(x)) = f(∛(x – 1) + 4) = ((∛(x – 1) + 4) – 4)³ + 1 = (∛(x – 1))³ + 1 = x
f⁻¹(f(x)) = f⁻¹((x – 4)³ + 1) = ∛(((x – 4)³ + 1) – 1) + 4 = (x – 4) + 4 = x
Therefore, f and f⁻¹ are inverse functions.
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