One way the function can be written is as y=f(x)=a*b^{x} and the goal is to find a and b. Since f(1)=1.5 and f(-1)=6, we get the following system of two equations and two unknowns: 1.5=ab, 6=a/b. One of the many ways to solve this system is to divide the second equation by the first to get 4=6/1.5=(a/b)/(ab)=b^{-2} so that b^{2}=\frac{1}{4} an b=\frac{1}{2}. This then implies that a=6b=6* 1/2=3 and the answer is y=3*(1/2)^{x}=3*2^{-x}.
Another way this type of problem is often solved is to write the function as y=f(x)=a*e^{kx} and the goal is to find a and k. The same points as above give the following system of equations: 1.5=ae^{k}, 6=ae^{-k}. Dividing the second equation by the first results in 4=6/1.5=(ae^{-k})/(ae^{k})=e^{-2k} so that -2k=ln(4) and k=-1/2*ln(4)=ln(4^{-1/2})=ln(0.5)\approx -0.693147. Then a=6e^{k}=6e^{ln(0.5)}=6*0.5=3 and the answer can be written in approximate form as y=f(x)\approx 3e^{-0.693147x}.
