Answer
$$f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \sqrt x ;\,\,\,\,\,\,\,\left[ {4,9} \right] \cr
& {\text{The function f is a probability density function of a random variable X in the interval }} \cr
& \left[ {a,b} \right]{\text{ if}} \cr
& 1{\text{ condition}}:f\left( x \right) \geqslant 0{\text{ for all }}x{\text{ in the interval }}\left[ {a,b} \right].{\text{ then}} \cr
& \sqrt x {\text{ is positive for all real number }}x \geqslant 0,{\text{ then }}f\left( x \right) \geqslant 0{\text{ for the given interval}} \cr
& \cr
& 2{\text{ condition}}:\int_a^b {f\left( x \right)} dx = 1.{\text{ then}} \cr
& \int_4^9 {\sqrt x } dx \cr
& \int_4^9 {{x^{1/2}}} dx \cr
& {\text{integrating}} \cr
& = \left( {\frac{2}{3}{x^{3/2}}} \right)_4^9 \cr
& = \frac{2}{3}\left( {{9^{3/2}} - {4^{3/2}}} \right) \cr
& {\text{simplifying}} \cr
& = \frac{2}{3}\left( {27 - 8} \right) \cr
& = \frac{{38}}{3} \cr
& \cr
& \int_a^b {f\left( x \right)} dx \ne 1 \cr
& \cr
& {\text{The condition 2 is not satisfied}}{\text{, then }} \cr
& f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr} $$
