Question

Let f: R→R and g: R→R be the functions given below. Find the compositions (f∘g)(x) and (g∘f)(x), and their domains, and provide the simplified expressions: a) Composition (f∘g)(x): (f∘g)(x)=f(g(x)) b) Composition (g∘f)(x): (g∘f)(x) = g(f(x))

Answer 1

Let f: R→R and g: R→R be the functions

a)
`(f \circ g)(x)` = f(g(x))

b)
`(g \circ f)(x)` = g(f(x))

Step-by-step explanation:

In finding the composition
`\( (f \circ g)(x) \)`, we substitute the function g(x) into f(x). This means that the output of g(x) serves as the input for f(x). Similarly, for
`\( (g \circ f)(x) \)`, the output of f(x) becomes the input for g(x).

For
`\( (f \circ g)(x) \)`, we begin by computing g(x) according to its given expression and then use this result as the input for f(x). This sequential operation allows us to find the composed function
`\( (f \circ g)(x) \)`

On the other hand, when determining
`\( (g \circ f)(x) \)`, we first find f(x) using its given expression and then input this outcome intog(x) to obtain the composed function
`\( (g \circ f)(x) \)`.

It's crucial to recognize that the domain of the composed functions might be restricted by the domains of the individual functions involved. In some cases, certain values might not be permissible due to restrictions posed by the original functions' domains, impacting the domain of the composed function accordingly.