Solving Functions

Question

The number of bacteria in a refrigerated food product is given by , where  is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by  , where  is the time in hours.Find the composite function : 

Answer 1

The "Composite Function N(T(t))" means that we are trying to evaluate the function N with the function T(t) as the input. In other words, we are plugging one function into another. We can plug the function T(t) into the function N just as we would plug any other value in, such as 1 or 2 or 3, etc. Then we just apply the function N to the (2t + 1.2) , and simplify the result. Here's what the process looks like:

Plug (2t + 1.2) into N: N(2t+1.2) = 28(2t+1.2)² - 98(2t+1.2) + 36

See how the (2t+1.2) is used wherever there was a "T" in the original N(T) function? Now we simplify:

First Term: (2t+1.2)² = 4t² + 4.8t + 1.44, so multiply this by 28 and you get 112t² + 134.4t + 40.32

Second Term: 98(2t+1.2) = 196t + 117.6 (just use the distributive property to multiply the 98 through)

Last Term: this one is just the constant 36

So, our final answer is what we get when we collect all of the "like terms" together from all three of the terms mentioned above. This gives us:

112t² + (134.4 + 196)t + (40.32 + 117.6 + 36) Simplify a bit more and we finish with:

112t² + 330.4t + 193.92

I hope this helps make things clearer!

-Taylor K.