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Algebra

Functions

Function Problem Solving

         

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

Details and assumptions

  • f−1(x) f^{-1}(x) denotes the inverse function of f(x). f(x).
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For three functions ff, gg and hh f(x)=x−3,(h∘g)(x)=6x+4.f(x)=x-3, (h \circ g)(x)=6x+4. What is the value of xx that satisfies (h∘(g∘f))(x)=52?(h \circ (g \circ f))(x)=52?

Details and assumptions

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

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

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If  X={−5,0,5}\ X = \{ -5, 0, 5 \} and Y={y∣−11≤y≤8, y∈Z} Y = \{ y \mid -11 \leq y \leq 8,\ y \in \mathbf{Z} \} , how many functions f:X→Y f: X \rightarrow Y are there such that x⋅f(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.

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

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