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Question

# If f(x) defines a constant function over $(-\infty, \infty)$, how many elements are in the range of $(f \circ f)(x)$?

Solution

Verified

Knowing that $f(x)$ be a constant function:

$\text{Let}~f(x)=c$

Where $c$ is a constant, we will find $(f\cdot f)(x)$:

\begin{aligned} (f\cdot f)(x)&=f(f(x))\\ &=f(c) \end{aligned}

In here $c$ is only one element in ranged of $(f\cdot f)(x)$. Therefore, there will be single element in range of $(f\cdot f)(x)$ if f(x) be a constant.

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