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 \(h(g(x))\).
For two sets \[X=\{-2, -1, 0, 1, 2\}, Y=\{y \mid y \text{ is an integer}\},\] function \(f: X \to Y\) is defined as \[f(x)=\begin{cases} x+11 & \text{ if } x > 0 \\ -x^2+4 & \text{ if } x \leq 0. \end{cases} \] What is the sum of all the elements of the range of \(f\)?
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.
Consider two functions \[f(x)=ax+1, g(x)=-x-2.\] If it always holds that \(f\circ g=g\circ f\), what is the value of the constant \(a\)?