Given f(x) = 7x³, to find f¹(x) which represents the inverse function, we need to swap the x and y variables and solve for y.
1. Start with the equation f(x) = 7x³
2. Replace f(x) with y: y = 7x³
3. Swap x and y to get x = 7y³
4. Solve for y by taking the cube root of both sides: y = (x/7)^(1/3)
Now, to determine if f¹(x) is a function, we need to check if for every x-value in the domain of f(x), there exists a unique y-value in the range of f(x).
In this case, f¹(x) = (x/7)^(1/3) is a function because for every x in its domain (all real numbers), there is a unique y-value associated with it. Each x-value corresponds to only one y-value, satisfying the definition of a function.
Therefore, the correct answer is:
f(x) is a function.
