One way to find the inverse function is to exchange the roles of x and y, and then solve for the new y. This works because the graph of the inverse of a function is the reflection of the graph of the function over the line y=x.

So ` ``f(x)=7x^3+4 ==> y=7x^3+4 `

Exchanging x and y we get:

`x=7y^3+4 `

Now solve for y:

`7y^3=x-4 `

`y^3=(x-4)/7 `

`y=root(3)((x-4)/7) `

So `f^(-1)(x)=root(3)((x-4)/7) `

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Note that the original function took an input x, cubed it, multiplied the result by 7, and then added 4 to the result. The inverse function undoes these operations in the reverse order (much as you put on socks then shoes, then take off the shoes then the socks.) So the inverse function should take an input x, subtract 4, divide the result by 7, and finally take the cubed root of the result.

We can verify the result by seeing that `f(f^(-1)(x))=f^(-1)(f(x))=x ` , and we can see that the graphs of the function and its inverse are reflections over the line y=x:

To find the inverse function of F(x)=7x^3+4 you have to first solve in terms of x. Keep in mind that f(x) is the same as y.

y = 7x^3 + 4

y - 4 = 7x^3

(y - 4)/7 = x^3

x = ((y - 4)/7)^1/3

Now you can swap the x and y and your inverse function becomes

f^-1(x) = ((x - 4)/7)^1/3

The function f(x) = 7x^3+4.

To determine the inverse function `f^-1(x)` , rewrite y = 7*x^3 + 4 in a form where x is expressed in terms of y.

`y = 7*x^3 + 4`

`(y - 4) = 7*x^3`

`(y - 4)/7 = x^3`

`x = ((y - 4)/7)^(1/3)`

Now interchange x and y, `y = ((x - 4)/7)^(1/3)`

This gives the inverse function of `f(x) =7x^3+4` as `f^-1(x) = ((x - 4)/7)^(1/3)`