How do I find the inverse of the function f(x) = ∛(x/7) + 9?

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To find the inverse of the function f(x) = ∛(x/7) + 9, we will follow these steps:

  1. Replace f(x) with y: Start by writing the equation as:

    2. y = ∛(x/7) + 9

  2. Isolate the cube root: To isolate the cube root term, we will subtract 9 from both sides:

    3. y – 9 = ∛(x/7)

  3. Cube both sides: Next, we will eliminate the cube root by cubing both sides:

    4. (y – 9)3 = x/7

  4. Multiply by 7: To solve for x, we then multiply both sides by 7:

    5. x = 7(y – 9)3

  5. Switch x and y: Now, we replace y with x to find the inverse function:

    6. f-1(x) = 7(x – 9)3

  6. Final result: Therefore, the inverse function is:

    7. f-1(x) = 7(x – 9)3

In summary, to find the inverse of f(x) = ∛(x/7) + 9, we followed these steps to isolate x, cube to eliminate the root, and finally switch our variables to express the inverse function.

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