Function Problem Solving


If f(x)=3x3 f(x) = 3x - 3 and g(x)=3x+1 g(x) = -3x + 1 , what is the value of a a satisfying (gf1)(a)=4? \left(g \circ f^{-1}\right)(a) = 4?

Details and assumptions

  • f1(x) f^{-1}(x) denotes the inverse function of f(x). f(x).

For three functions ff, gg and hh f(x)=x3,(hg)(x)=6x+4.f(x)=x-3, (h \circ g)(x)=6x+4. What is the value of xx that satisfies (h(gf))(x)=52?(h \circ (g \circ f))(x)=52?

Details and assumptions

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

For two sets X={2,1,0,1,2},Y={yy is an integer},X=\{-2, -1, 0, 1, 2\}, Y=\{y \mid y \text{ is an integer}\}, function f:XYf: X \to Y is defined as f(x)={x+11 if x>0x2+4 if x0.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 ff?

If  X={5,0,5}\ X = \{ -5, 0, 5 \} and Y={y11y8, yZ} Y = \{ y \mid -11 \leq y \leq 8,\ y \in \mathbf{Z} \} , how many functions f:XY f: X \rightarrow Y are there such that xf(x) x \cdot f(x) is a constant function for all elements of xx in XX?

Details and assumptions

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

Consider two functions f(x)=ax+1,g(x)=x2.f(x)=ax+1, g(x)=-x-2. If it always holds that fg=gff\circ g=g\circ f, what is the value of the constant aa?


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