The number of bacteria (b) in a batch of refrigerated food is given by the function b(t) = 20t2 − 70t + 300, where t is the temp
erature of the food in degrees Fahrenheit. The function t(h) = 2h + 3 gives the temperature of the food h hours after it is removed from refrigeration. Which expression correctly represents the number of bacteria (b) in the food as a function of the number of hours (h) the food is unrefrigerated?
We want to combine b(t) and t(h) to create a function that described the number of bacteria, b, as a function of time, h. This function can be called b(t(h)), or b(h). Consider the function b(t), or bacteria as a function of temperature. b(t) = 20t² - 70t +300 This describes the number of bacteria, given the temperature.
t(h) described the temperature as a function of time, specifically hours after refrigeration: t(h) = 2h + 3
Since the time h can tell us the temperature t, and the temperature t can tell us the # of bacteria b, we can create a function that tells us the number of bacteria, b, given hours following refrigeration, h.
To do this, we plug t(h) in for every t in the b(t) function: b(t(h)) = 20 (2h+3)² - 70(2h+3) + 300 We can also call this function b(h), since we now can express b as a function of h.
