Questions: Graphs and Functions Expressing a function as a composition of two functions Suppose H(x)=(2x-9)^3. 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.) f(x)= g(x)=

Graphs and Functions
Expressing a function as a composition of two functions

Suppose H(x)=(2x-9)^3.
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.)

f(x)=

g(x)=

Transcript text: Graphs and Functions Expressing a function as a composition of two functions Suppose $H(x)=(2 x-9)^{3}$. 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.) \[ f(x)= \] $\square$ \[ g(x)= \] $\square$ $\square$

Solution

Solution Steps

To express $ H(x) = (2x - 9)^3 $ as a composition of two functions $ f $ and $ g $, we need to identify two functions such that $ (f \circ g)(x) = f(g(x)) = H(x) $. One way to do this is to let $ g(x) $ be the inner function and $ f(x) $ be the outer function. For example, we can let $ g(x) = 2x - 9 $ and $ f(x) = x^3 $. This way, $ f(g(x)) = (2x - 9)^3 $.

Step 1: Define the Inner Function $ g(x) $

To express $ H(x) = (2x - 9)^3 $ as a composition of two functions, we start by defining the inner function $ g(x) $. Let: \[ g(x) = 2x - 9 \]

Step 2: Define the Outer Function $ f(x) $

Next, we define the outer function $ f(x) $ such that when $ g(x) $ is substituted into $ f(x) $, we get $ H(x) $. Let: \[ f(x) = x^3 \]

Step 3: Verify the Composition

To verify that $ (f \circ g)(x) = H(x) $, we substitute $ g(x) $ into $ f(x) $: \[ (f \circ g)(x) = f(g(x)) = f(2x - 9) = (2x - 9)^3 \]