18. $$ f(x)=x^{3}+5, \quad g(x)=\sqrt[3]{x-5} $$

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We are given the functions

f(x) = x³ + 5  
  g(x) = ∛(x − 5)

To show that these two functions are inverses of each other, we need to check that:

f(g(x)) = x  and  g(f(x)) = x

Step 1. Compute f(g(x)):
  f(g(x)) = (g(x))³ + 5  
Since g(x) = ∛(x − 5), raising it to the power of 3 gives:
  (∛(x − 5))³ = x − 5  
Thus,
  f(g(x)) = (x − 5) + 5 = x

Step 2. Compute g(f(x)):
  g(f(x)) = ∛(f(x) − 5)  
Substitute f(x) = x³ + 5:
  g(f(x)) = ∛((x³ + 5) − 5) = ∛(x³)
Since ∛(x³) = x for all real x:
  g(f(x)) = x

Since both composite functions yield x, we conclude that g is the inverse of f and vice versa.
$$ f(x) = x^3 + 5 $$ and $$ g(x) = \sqrt[3]{x - 5} $$ are inverses of each other because $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$.

18. \( f(x)=x^{3}+5, \quad g(x)=\sqrt[3]{x-5} \)

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