$$f(x):=\begin{cases} [x] & -2≤ x ≤ -1 \\ |x| +1 & -1≤ x ≤ 2 \end{cases}$$

$$ g(x):=\begin{cases} [x] & -π ≤ x ≤ 0 \\ \sin(x) & \phantom{+}0 ≤ x ≤ π \end{cases}$$

Range of $g\circ f(x)$ ( $[x]$ represent greater integer function).

In this question I draw the operator diagram and find the range of individual composite function . I want to know that I have to take union or intersection of the range.

  • $\begingroup$ Your title doesn't mirror the question : it should be something like "Range of the composition of two functions" $\endgroup$
    – Jean Marie
    Jun 8, 2016 at 6:27
  • $\begingroup$ In general, the range of the compostion is neither the intersection, nor the union of the ranges. It is, however, a subset of the range of the last function in the compostion (in your case $g$). $\endgroup$ Jun 8, 2016 at 6:37
  • $\begingroup$ $f(x)$'s range will be domain for $g(x)$ and using that domain get the range for $g(x)$ , and if range of $f(x)$ lies outside domain of $g(x)$ $g(f(x))$ won't be defined. i would suggest analyze this using graph to get better feel $\endgroup$
    – avz2611
    Jun 8, 2016 at 6:40
  • $\begingroup$ I want to know what I should do to two answer which are coming from the given equation . $\endgroup$ Jun 8, 2016 at 6:42

1 Answer 1


The range of $f$ consists of the two points $-2,-1$ and the interval $[1,3]$. That lies entirely within the domain of $g$. Hence the range of $g\circ f$ consists of the two points $-2,-1$ and the interval $[k,1]$ where $k=\sin3\approx0.1411$.


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