Decompose the function y=e
√
x
in the form y=f(u) and u=g(x). Select the correct functions for y=f(u) and u=g(x).

Question

Decompose the function y=e
√
x
 in the form y=f(u) and u=g(x).  Select the correct functions for y=f(u) and u=g(x).

BlackTom AI · Accepted Answer

The task asks to decompose y = e^{√x} into the form y = f(u) with y depending on an intermediate variable u, where u = g(x).
Option 1: y = e^u. Here, if we choose u to be √x, then y = e^{√x} becomes y = e^u, which correctly expresses the original function in terms of u. This is a valid f(u) because the exponential function is applied to u directly.
Option 2: u = x. If we take u = x, then y = f(u) would become y = e^x, since y = f(u) with f(u) = e^u. But e^x is not the same as e^{√x} for general x, so this choice fails to reproduce the original y. It mis-specifies the inner function g(x) and therefore does not provide a correct decomposition.
Overall, the correct decomposition requires choosing u to capture the inner expression √x. Since the given option set includes y = e^u (which is compatible with u = √x) but u = x (which makes the outer function operate on x rather than on √x), only the first option aligns with a correct decomposition when paired with the appropriate g(x). Note that in a complete list, the intended pair would be y = e^u and u = √x, which is missing here; the provided u = x does not yield the original y after substitution.

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 3 - Trig, Exponential and Chain Rule - PreClass

Decompose the function y=e √ x in the form y=f(u) and u=g(x). Select the correct functions for y=f(u) and u=g(x).

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