To determine if the functions \( f(x) = 5x + 1 \) and \( g(x) = x + 5 \) are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).
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Calculate \( f(g(x)) \):
\[ g(x) = x + 5 \] Now substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x + 5) = 5(x + 5) + 1 = 5x + 25 + 1 = 5x + 26 \]
Since \( f(g(x)) = 5x + 26 \neq x \), the functions \( f \) and \( g \) do not satisfy the first condition for being inverse functions.
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Calculate \( g(f(x)) \):
Now let's calculate \( g(f(x)) \):
\[ f(x) = 5x + 1 \] Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(5x + 1) = (5x + 1) + 5 = 5x + 1 + 5 = 5x + 6 \]
Since \( g(f(x)) = 5x + 6 \neq x \), the functions \( f \) and \( g \) fail the second condition as well.
Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \) holds true, we conclude that \( f(x) \) and \( g(x) \) are not inverse functions.