Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g .

h(x) = √ (x⁴ + 2 )

Guida verificata passo dopo passo

Step 1: Understand the problem. We need to express the function $ h(x) = \sqrt{x^4 + 2} $ as a composition of two functions $ f $ and $ g $, such that $ h(x) = (f \circ g)(x) = f(g(x)) $.

Step 2: Identify the inner function $ g(x) $. Look for a part of $ h(x) $ that can be isolated as a simpler function. In this case, consider $ g(x) = x^4 + 2 $.

Step 3: Determine the outer function $ f(x) $. Since $ g(x) = x^4 + 2 $, the remaining operation in $ h(x) $ is taking the square root. Therefore, let $ f(x) = \sqrt{x} $.

Step 4: Verify the composition. Substitute $ g(x) $ into $ f(x) $ to ensure it reconstructs $ h(x) $: $ f(g(x)) = f(x^4 + 2) = \sqrt{x^4 + 2} $, which matches $ h(x) $.

Step 5: Conclude that the functions $ f(x) = \sqrt{x} $ and $ g(x) = x^4 + 2 $ are valid choices for the outer and inner functions, respectively, such that $ h(x) = f(g(x)) $.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (f o g)(x) = f(g(x)), where f is the outer function and g is the inner function. Understanding how to decompose a function into its components is essential for identifying suitable outer and inner functions that yield the desired composite function.

Video consigliato:

3:48

Evaluate Composite Functions - Special Cases

Function Decomposition

Function decomposition involves breaking down a complex function into simpler constituent functions. This process is crucial when working with composite functions, as it allows us to identify potential candidates for the inner and outer functions. For example, in the function h(x) = √(x⁴ + 2), recognizing the structure of the expression can guide us in selecting appropriate f and g.

Video consigliato:

Percorso guidato

06:21

Properties of Functions

Square Root Function

The square root function, denoted as √x, is a fundamental mathematical function that returns the non-negative value whose square equals x. In the context of composite functions, it often serves as an outer function. Understanding its properties, such as its domain and range, is vital for determining how it can be combined with other functions to form a composite function like h(x).

Video consigliato:

Percorso guidato