Algebra

# Function Problem Solving

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 $$a$$?

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