Question

Write the composite function in the form f(g(x)). [identify the inner function u = g(x) and the outer function y = f(u).] (use non-identity functions for f(u) and g(x).) y = e⁷ˣ (f(u), g(x)) =

a) y = e⁽ᵘ⁷⁾
b) y = ln(7eˣ)
c) y = e⁷ˣ
d) y = 7eˣ

Answer 1

The composite function y = ln(7e^x) can be expressed as f(g(x)) by identifying the inner function g(x) as x and the outer function f(u) as ln(7u). Using the property of logarithms that separates the log of a product, this simplifies to ln(7) + x, showing how natural log and exponential functions are inverses of each other.

Step-by-step explanation:

The question involves creating a composite function in the form of f(g(x)). To identify the inner and outer functions, we will look at the given expressions:

• b) y = ln(7e^x)

For expression b), we will express this as a composite function. The inner function g(x) is simply x since e is a constant and doesn't affect the selection of the inner function. The outer function f(u) deals with the ln operation and we can consider it as taking the natural logarithm of its input u which would be the result of the inner function g(x) times 7.

Therefore, using properties of logarithms:

y = f(g(x)) = f(u) = ln(7g(x)) = ln(7) + ln(e^x) = ln(7) + x

This simplifies to the sum of ln(7) and x, which shows that taking an exponential function and its inverse, the natural log (ln), are ways to 'undo' each other, as they are inverse functions.