Can you explain to us if we have the functions c(t) = 3t + 2, where c is the cost in dollars of t hours of tutoring and t(m) = m2 - 1 where t is hours of tutoring and m is the number of minutes tutored, what the input and output values would be of the composite function c(t(m))?
To find the input and output values of the composite function c(t(m)), we need to substitute the function t(m) into the function c(t).
The function t(m) = m^2 - 1 represents the number of hours of tutoring (t) in terms of the number of minutes tutored (m).
Substituting this expression for t in the function c(t), we get:
c(t(m)) = c(m^2 - 1)
Now let's simplify this expression. The original function c(t) = 3t + 2 represents the cost in dollars (c) of t hours of tutoring.
Substituting t(m) into the function c(t), we get:
c(m^2 - 1) = 3(m^2 - 1) + 2
Expanding this expression further:
c(m^2 - 1) = 3m^2 - 3 + 2
c(m^2 - 1) = 3m^2 - 1
Therefore, the composite function c(t(m)) is given by c(m^2 - 1) = 3m^2 - 1.
Now, let's talk about the input and output values of this composite function.
- Input values: In this case, the input values of the composite function c(t(m)) are the number of minutes tutored (m).
- Output values: The output values of the composite function c(t(m)) are the cost in dollars (c) of tutoring, which is computed using the formula c(m^2 - 1) = 3m^2 - 1.
So, by substituting different values for m, you can find the corresponding cost in dollars for the given number of minutes tutored.