Redo problem 13 from section 1.8 of your textbook (page 62), but use the functions in the questions below rather than the functions in your textbook:
(a) In order to express the function y = 1311 x − 4, as a composition of two functions the outer of which is an exponential function,
we would let u be equal to the inner function, u = ,
and then write y as a function of u, as follows: y = .
(b) In order to express the function P = √{ 20 t5 + 1 }, as a composition of two functions the outer of which is a root function,
we would let u be equal to the inner function, u = ,
and then write P as a function of u, as follows: P = .
(c) In order to express the function w = 5 ln( −1 r− 10 ) as a composition of two functions the outer of which is a logarithmic function,
we would let u be equal to the inner function, u = ,
and then write w as a function of u, as follows: w = .
We don't have your textbook.
(a) To express the function y = 1311x - 4 as a composition of two functions, the outer of which is an exponential function, we need to let u be equal to the inner function.
Let u = 1311x - 4.
Now, to express y as a function of u, we can use an exponential function.
Let f(u) = e^u, where e is Euler's number.
Therefore, y = f(u) = e^(1311x - 4).
So, the composition of two functions that expresses y = 1311x - 4, with the outer function being exponential, is y = e^(1311x - 4).
(b) To express the function P = √{ 20t^5 + 1 } as a composition of two functions, the outer of which is a root function, we need to let u be equal to the inner function.
Let u = 20t^5 + 1.
Now, to express P as a function of u, we can use a root function.
Let g(u) = √u.
Therefore, P = g(u) = √(20t^5 + 1).
So, the composition of two functions that expresses P = √{ 20t^5 + 1 }, with the outer function being a root function, is P = √(20t^5 + 1).
(c) To express the function w = 5ln(-1/r - 10) as a composition of two functions, the outer of which is a logarithmic function, we need to let u be equal to the inner function.
Let u = -1/r - 10.
Now, to express w as a function of u, we can use a logarithmic function.
Let h(u) = ln(u), where ln represents the natural logarithm.
Therefore, w = 5h(u) = 5ln(-1/r - 10).
So, the composition of two functions that expresses w = 5ln(-1/r - 10), with the outer function being a logarithmic function, is w = 5ln(-1/r - 10).