given the following function: f(x)={( kx 0<X<10 k(20-x) 10<X<20@0 otherwise)┤ Find the value of k such that f(x) is a probability density function.

Since f(x) is a probability density function, then:
∫_(-∞)^∞〖f(x)dx=〗 1
∴1=∫_0^10〖kx dx〗+k∫_10^20〖(20-x)〗 dx
=k[1/2 x^2 ]_0^10+k[20x-1/2 x^2 ]_10^20
=50k+k[400-200-20+50]=100k→k=1/100
Thus:
f(x)={(  x/100               0<X<10@((20-x))/100          10<X<20@0                      otherwise)┤

Its graph is: