Math Problem Statement

If the domain of f(x) is 1-4 (inclusive) and g(x) is 1-2 (inclusive) what is the domain of fg(x) and if gf(x)

Solution

To find the domains of (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x), we need to understand how function composition works with respect to domains.

1. For (fg)(x)(f \circ g)(x):

This is the composition of functions f(x)f(x) and g(x)g(x), which means that we apply g(x)g(x) first and then apply ff to the result of g(x)g(x). So, f(g(x))f(g(x)) is only valid when:

  • The output of g(x)g(x) lies within the domain of ff.

Given:

  • The domain of f(x)f(x) is [1,4][1, 4].
  • The domain of g(x)g(x) is [1,2][1, 2].

For (fg)(x)(f \circ g)(x) to be valid, we need the output of g(x)g(x) to fall within [1,4][1, 4]. Since g(x)g(x) is defined for [1,2][1, 2], we need to check if all outputs of g(x)g(x) (which lie between 1 and 2) are also within f(x)f(x)'s domain.

Thus, the domain of (fg)(x)(f \circ g)(x) is the set of values of xx such that g(x)g(x) lies within [1,4][1, 4]. Since g(x)g(x)'s outputs are always in [1,2][1, 2], and f(x)f(x) is defined for [1,4][1, 4], the domain of (fg)(x)(f \circ g)(x) is the domain of g(x)g(x), which is [1,2][1, 2].

2. For (gf)(x)(g \circ f)(x):

This is the composition g(f(x))g(f(x)), where we first apply f(x)f(x) and then apply gg to the result. For g(f(x))g(f(x)) to be valid:

  • The output of f(x)f(x) must lie within the domain of g(x)g(x).

Given:

  • The domain of f(x)f(x) is [1,4][1, 4].
  • The domain of g(x)g(x) is [1,2][1, 2].

For g(f(x))g(f(x)) to be valid, the output of f(x)f(x) must lie within [1,2][1, 2]. Since f(x)f(x) has a domain of [1,4][1, 4], we need to find which values of xx produce outputs within [1,2][1, 2]. Therefore, we must check where f(x)[1,2]f(x) \in [1, 2].

The domain of (gf)(x)(g \circ f)(x) is the set of values of xx such that f(x)[1,2]f(x) \in [1, 2]. We do not have explicit details about the function f(x)f(x), but we can infer that xx values for which f(x)f(x) lies in [1,2][1, 2] will give us the domain of (gf)(x)(g \circ f)(x). Thus, the domain of (gf)(x)(g \circ f)(x) is a subset of [1,4][1, 4], specifically where f(x)[1,2]f(x) \in [1, 2].

Summary:

  • The domain of (fg)(x)(f \circ g)(x) is [1,2][1, 2].
  • The domain of (gf)(x)(g \circ f)(x) is a subset of [1,4][1, 4], specifically where f(x)[1,2]f(x) \in [1, 2].

Would you like to explore further or clarify any points? Here are a few related questions:

  1. How does the range of a function affect its composition?
  2. Can you find the domain of a composition if the functions are given explicitly?
  3. What happens if the domain of one function does not overlap with the range of another in composition?
  4. How can function graphs help in visualizing composition domains?
  5. Can a function be composed with itself (i.e., f(f(x))f(f(x)))? What would the domain be?

Tip: Always remember to check that the range of the inner function fits within the domain of the outer function when composing them.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Domain of Functions

Formulas

Domain of (f ∘ g)(x) = Domain of g(x)
Domain of (g ∘ f)(x) = {x | f(x) ∈ Domain of g(x)}

Theorems

Composition of functions

Suitable Grade Level

Grades 9-11