Math Problem Statement

Determine the domains of the following functions based on the given functions: f(x) = 4/|x| and g(x) = x - 3.

Solution

Let's break down the problem based on the image provided:

The given functions are:

  • f(x)=4xf(x) = \frac{4}{|x|}
  • g(x)=x3g(x) = x - 3

You are asked to determine the domains of the following: (a) The domain of f(x)f(x). (b) The domain of g(x)g(x). (c) The domain of fg(x)f \circ g(x), which is the composition f(g(x))f(g(x)).

(a) Domain of f(x)=4xf(x) = \frac{4}{|x|}:

  • The function f(x)f(x) involves an absolute value in the denominator. For 4x\frac{4}{|x|} to be defined, the denominator cannot be zero.
  • Thus, x0x \neq 0.
  • The domain of f(x)f(x) is all real numbers except zero: (,0)(0,)(-\infty, 0) \cup (0, \infty).

(b) Domain of g(x)=x3g(x) = x - 3:

  • g(x)g(x) is a linear function with no restrictions on xx. It is defined for all real numbers.
  • Therefore, the domain of g(x)g(x) is (,)(-\infty, \infty).

(c) Domain of fg(x)=f(g(x))f \circ g(x) = f(g(x)):

  • To find the domain of the composition f(g(x))f(g(x)), first substitute g(x)=x3g(x) = x - 3 into f(x)f(x): f(g(x))=4x3f(g(x)) = \frac{4}{|x - 3|}
  • Similar to f(x)f(x), the denominator x3|x - 3| cannot be zero. This means x30x - 3 \neq 0, or x3x \neq 3.
  • Thus, the domain of f(g(x))f(g(x)) is all real numbers except x=3x = 3: (,3)(3,)(-\infty, 3) \cup (3, \infty).

Final Answers:

  • (a) Domain of f(x)f(x): (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • (b) Domain of g(x)g(x): (,)(-\infty, \infty)
  • (c) Domain of fg(x)f \circ g(x): (,3)(3,)(-\infty, 3) \cup (3, \infty)

Would you like more details on any of these points?


Here are 5 related questions to expand on this:

  1. What would happen to the domain of f(x)f(x) if we changed it to f(x)=4x2f(x) = \frac{4}{x^2}?
  2. How would the domain change if g(x)=1x3g(x) = \frac{1}{x-3}?
  3. How does the composition of two functions generally affect the domain?
  4. Could there be a case where f(g(x))f(g(x)) is undefined for an interval of values instead of just specific points?
  5. What is the domain of f(g(x))f(g(x)) if f(x)=1x+2f(x) = \frac{1}{x+2} and g(x)=1xg(x) = \frac{1}{x}?

Tip: When dealing with rational functions, always check for points where the denominator equals zero to determine domain restrictions.

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Math Problem Analysis

Mathematical Concepts

Function Domains
Absolute Value Functions
Composition of Functions

Formulas

f(x) = 4 / |x|
g(x) = x - 3
f(g(x)) = 4 / |x - 3|

Theorems

Domain of a function
Properties of absolute value functions
Function composition

Suitable Grade Level

Grade 11-12 or college-level mathematics