Math Problem Statement
Solution
Let's break down the problem based on the image provided:
The given functions are:
You are asked to determine the domains of the following: (a) The domain of . (b) The domain of . (c) The domain of , which is the composition .
(a) Domain of :
- The function involves an absolute value in the denominator. For to be defined, the denominator cannot be zero.
- Thus, .
- The domain of is all real numbers except zero: .
(b) Domain of :
- is a linear function with no restrictions on . It is defined for all real numbers.
- Therefore, the domain of is .
(c) Domain of :
- To find the domain of the composition , first substitute into :
- Similar to , the denominator cannot be zero. This means , or .
- Thus, the domain of is all real numbers except : .
Final Answers:
- (a) Domain of :
- (b) Domain of :
- (c) Domain of :
Would you like more details on any of these points?
Here are 5 related questions to expand on this:
- What would happen to the domain of if we changed it to ?
- How would the domain change if ?
- How does the composition of two functions generally affect the domain?
- Could there be a case where is undefined for an interval of values instead of just specific points?
- What is the domain of if and ?
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
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