Questions: Given functions f and g, find (a)(f ∘ g)(x) and its domain, and (b)(g ∘ f)(x) and its domain. f(x)=x^3, g(x)=x^2+3x-9

Transcript text: Given functions $f$ and $g$, find $(a)(f \circ g)(x)$ and its domain, and $(b)(g \circ f)(x)$ and its domain. \[ f(x)=x^{3}, g(x)=x^{2}+3 x-9 \]

Solution

Solution Steps

To solve the problem, we need to find the composition of the functions $ f $ and $ g $. The composition $ (f \circ g)(x) $ means we substitute $ g(x) $ into $ f(x) $. Similarly, $ (g \circ f)(x) $ means we substitute $ f(x) $ into $ g(x) $. The domain of each composition is determined by the domain of the inner function and the resulting expression.

Solution Approach

For $ (f \circ g)(x) $, substitute $ g(x) $ into $ f(x) $.
For $ (g \circ f)(x) $, substitute $ f(x) $ into $ g(x) $.
Determine the domain of each composition by considering the domain of the inner function and any restrictions from the resulting expression.

Step 1: Determine $ (f \circ g)(x) $

To find $ (f \circ g)(x) $, substitute $ g(x) = x^2 + 3x - 9 $ into $ f(x) = x^3 $. This gives: \[ (f \circ g)(x) = (x^2 + 3x - 9)^3 \]

Step 2: Determine the Domain of $ (f \circ g)(x) $

The domain of $ (f \circ g)(x) $ is determined by the domain of $ g(x) $, which is all real numbers, since $ g(x) = x^2 + 3x - 9 $ is a polynomial. Therefore, the domain of $ (f \circ g)(x) $ is all real numbers.

Step 3: Determine $ (g \circ f)(x) $

To find $ (g \circ f)(x) $, substitute $ f(x) = x^3 $ into $ g(x) = x^2 + 3x - 9 $. This gives: \[ (g \circ f)(x) = (x^3)^2 + 3(x^3) - 9 = x^6 + 3x^3 - 9 \]

Step 4: Determine the Domain of $ (g \circ f)(x) $

The domain of $ (g \circ f)(x) $ is determined by the domain of $ f(x) $, which is all real numbers, since $ f(x) = x^3 $ is a polynomial. Therefore, the domain of $ (g \circ f)(x) $ is all real numbers.

Final Answer

(a) $ (f \circ g)(x) = (x^2 + 3x - 9)^3 $ with domain all real numbers. \\ (b) $ (g \circ f)(x) = x^6 + 3x^3 - 9 $ with domain all real numbers. \\ \[ \boxed{(f \circ g)(x) = (x^2 + 3x - 9)^3, \text{ domain: all real numbers}} \] \[ \boxed{(g \circ f)(x) = x^6 + 3x^3 - 9, \text{ domain: all real numbers}} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!