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# Functions

Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.

If we have a real-valued function \( f(x) = x^2 + 27 \), the values of \( f(x) \) will necessarily fall in the following range:

\[ a \leq f(x). \]

What is the value of \( a \)?

Consider two sets: \( X = \{ a, b, c, d \}, \ Y = \{ 1, 2, 3 \}. \)

If a function is defined from \( X \) to \( Y \), what is the maximum possible number of such functions?

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