Questions: Suppose H(x)=(3x-4)^7. Find two functions f and g such that (f∘g)(x)=H(x). Neither function can be the identity function. (There may be more than one correct answer.)

Suppose H(x)=(3x-4)^7.
Find two functions f and g such that (f∘g)(x)=H(x).
Neither function can be the identity function.
(There may be more than one correct answer.)

Transcript text: ALEKS Chapter 11 Informative an... ALEKS - Aaron Price - Le... Graphs and Functions Expressing a function as a composition of two functions Suppose $H(x)=(3 x-4)^{7}$. Find two functions $f$ and $g$ such that $(f \circ g)(x)=H(x)$. Neither function can be the identity function. (There may be more than one correct answer.) $\square$ Explanation Check

Solution

Solution Steps

To express the function $ H(x) = (3x - 4)^7 $ as a composition of two functions $ f $ and $ g $, we need to identify an inner function $ g(x) $ and an outer function $ f(u) $ such that $ f(g(x)) = H(x) $. A common approach is to let $ g(x) $ be the expression inside the parentheses and $ f(u) $ be the operation applied to that expression. In this case, we can choose $ g(x) = 3x - 4 $ and $ f(u) = u^7 $.

Step 1: Define the Functions

We have the function $ H(x) = (3x - 4)^7 $. To express this as a composition of two functions, we define:

$ g(x) = 3x - 4 $
$ f(u) = u^7 $

Step 2: Composition of Functions

The composition of the functions is given by: \[ (f \circ g)(x) = f(g(x)) = f(3x - 4) = (3x - 4)^7 \] This confirms that our choice of $ f $ and $ g $ correctly represents $ H(x) $.

Step 3: Evaluate the Function

To evaluate $ H(x) $ at $ x = 2 $: