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 = �ã{ 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 = .
(a) To express the function y = 1311x - 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. Let u = 311x - 4.
To write y as a function of u, we need to find an exponential function that can be composed with u to obtain y. We can use the exponential function f(t) = e^t.
Let u = 311x - 4 and let v = e^u.
Now, substitute v into y to express y as a function of u:
y = 1311x - 4 = 1311(v) = 1311(e^u).
Therefore, the function y = 1311x - 4 can be expressed as a composition of two functions, where the outer function is an exponential function, y = 1311(e^u).
(b) To express the function P = √(20t^5 + 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. Let u = 20t^5 + 1.
To write P as a function of u, we need to find a root function that can be composed with u to obtain P. We can use the square root function f(x) = √x.
Let u = 20t^5 + 1 and let v = √u.
Now, substitute v into P to express P as a function of u:
P = √(20t^5 + 1) = √(v) = v = √(√u) = (√√u).
Therefore, the function P = √(20t^5 + 1) can be expressed as a composition of two functions, where the outer function is a root function, P = (√√u).
(c) 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. Let u = -1/r - 10.
To write w as a function of u, we need to find a logarithmic function that can be composed with u to obtain w. We can use the natural logarithm function f(x) = ln(x).
Let u = -1/r - 10 and let v = ln(u).
Now, substitute v into w to express w as a function of u:
w = 5 ln(-1/r - 10) = 5 ln(v) = 5 ln(ln(u)).
Therefore, the function w = 5 ln(-1/r - 10) can be expressed as a composition of two functions, where the outer function is a logarithmic function, w = 5 ln(ln(u)).