How can we verify that the function g(x) = 13x is the inverse of the function f(x) = 3x?

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To verify that g(x) is the inverse of f(x), we need to check two key conditions. In general, if g is the inverse of f, then the following two equations must hold:

  • f(g(x)) = x
  • g(f(x)) = x

Let’s calculate each of these:

1. Checking f(g(x))

First, we will substitute g(x) into f:

f(g(x)) = f(13x)

Now, since f(x) = 3x, we plug in 13x:

f(13x) = 3(13x) = 39x

As we can see, f(g(x)) = 39x, which does not equal x. Therefore, this condition fails.

2. Checking g(f(x))

Next, let’s substitute f(x) into g:

g(f(x)) = g(3x)

Using g(x) = 13x, we substitute 3x:

g(3x) = 13(3x) = 39x

Again, we find that g(f(x)) = 39x, which still does not equal x, indicating that this condition also fails.

Conclusion

Since neither f(g(x)) nor g(f(x)) equals x, we conclude that g(x) = 13x is not the inverse of f(x) = 3x.

Thus, the expression that could be used to verify that g(x) is the inverse of f(x) would be to check these two conditions of composition. In this case, both checks returned 39x instead of x, proving that they are not inverses of each other.

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