Working with composite functions Find possible choices for...

Working with composite functions Find possible choices for outer and inner functions $f$ and $g$ such that the given function h equals f\circ g. Give the domain of h.
$$h(x)=\frac{2}{\left(x^{6}+x^{2}+1\right)^{2}}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Composite Functions

Composite functions are built by applying one function to the result of another. In other words, if you have two functions, f and g, then the composite function f?g is defined by f(g(x)). This concept is useful for breaking down complicated expressions into simpler parts, which can then be analyzed and manipulated more easily.

Inner and Outer Functions

When decomposing a function into a composite form, the inside part of the expression is typically chosen as the inner function, g, while the remaining part that acts on g(x) is the outer function, f. This separation allows one to analyze the effect of each function independently and understand how changes in one part of the function affect the overall composition.

Domain of Functions

The domain of a function is the set of all input values for which the function is defined. In the context of composite functions, it is important to consider the domain of the inner function and ensure that the outputs of the inner function fall within the domain of the outer function. The overall domain of the composite function consists of those inputs that satisfy both these conditions.

Transcript

Please give Ace some feedback