Algebra
# Functions

If \( f(x) = 3x - 3 \) and \( g(x) = -3x + 1 \), what is the value of \( a \) satisfying \( \left(g \circ f^{-1}\right)(a) = 4? \)

**Details and assumptions**

- \( f^{-1}(x) \) denotes the inverse function of \( f(x). \)

For three functions \(f\), \(g\) and \(h\) \[f(x)=x-3, (h \circ g)(x)=6x+4.\] What is the value of \(x\) that satisfies \[(h \circ (g \circ f))(x)=52?\]

**Details and assumptions**

Composite function \((h \circ g)(x)\) denotes \(h(g(x))\).

If \(\ X = \{ -5, 0, 5 \} \) and \( Y = \{ y \mid -11 \leq y \leq 8,\ y \in \mathbf{Z} \} \), how many functions \( f: X \rightarrow Y \) are there such that \( x \cdot f(x) \) is a constant function for all elements of \(x\) in \(X\)?

**Details and assumptions**

\( \mathbf{Z} \) is the set of all integers.

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