A piecewise function is defined by p(x)= cases x+2, & x4

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

A piecewise function

p

is defined by

p(x)=\begin{cases} x+2, & x<-1,\\ \sqrt{4-x}, & -1\le x\le 4,\\ \dfrac{2}{x-4}, & x>4. \end{cases}

Another function is

q(t)=\sqrt{5t-t^{2}}

. Answer the following questions about the composite function

q\circ p

. (i) For the branch

x<-1

, find all

x

such that

q(p(x))

is real. (ii) Repeat part (i) for the branch

-1\le x\le 4

. (iii) Repeat part (i) for the branch

x>4

. (iv) Combine your results to write the complete domain of

q\circ p

and show that it can be expressed as the union of two disjoint intervals.