Is the Federal Reserve responsible for determining the inflation rate and the unemployment rate?

The Federal Reserve does not set the inflation or unemployment rates. These rates are naturally fluctuating based on market activities. Typically, as inflation rises, unemployment decreases and vice versa (except in the case of stagflation in 1970's). The Federal Reserve DOES, however, adjust interest rates and various other rates to control the money supply in order to combat unemployment and inflation. See the "Money Supply Theory."