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?
For two sets \[X=\{a,b,c\}, Y=\{7, 11, 13, 17, 20, 32\},\] \(f\) is an injective function from \(X\) to \(Y\). If \(f(a)=7\) and \(f(b)=17\), what is the sum of all the elements of \(Y\) that can possibly be the value of \(f(c)\)?
Given that the domain and codomain of the function \(y=\sqrt{169-x^2}\) are restricted to the real numbers, how many elements of the domain of \(y=\sqrt{169-x^2}\) are integers?
For two sets \[X=\{-1,1,a\}, Y=\{3,4, b\},\] \(f(x)=x^3+4\) is a bijective function from \(X\) to \(Y\). What is the value of \(a+b\)?