# Range when two piecewise functions are composed

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

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

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.

• Your title doesn't mirror the question : it should be something like "Range of the composition of two functions" Jun 8, 2016 at 6:27
• 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 $g$). Jun 8, 2016 at 6:37
• $f(x)$'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 Jun 8, 2016 at 6:40
• I want to know what I should do to two answer which are coming from the given equation . Jun 8, 2016 at 6:42

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 $k=\sin3\approx0.1411$.