Math Problem Statement
Solution
The problem in the image is asking to determine the domains of two functions and , as well as the domain of the composition of these functions .
Solution:
(a) Domain of
The domain of a function is the set of all input values for which the function is defined. For , the function is undefined when the denominator is zero. Therefore: The domain of is all real numbers except zero, which in interval notation is:
(b) Domain of
Similarly, for , the denominator is , and the function is undefined when , or equivalently when . Thus, the domain of is:
(c) Domain of
The composition means applying to the output of . First, , so: Now, is a linear function, which is defined for all real numbers except when , since is undefined at . Therefore, the domain of is also:
Summary:
- (a) Domain of :
- (b) Domain of :
- (c) Domain of :
Would you like further clarification or details on any part of the solution?
Here are 5 related questions to deepen your understanding:
- How do you determine the domain of a function involving fractions?
- What happens if you compose two functions with different domains?
- How do you graph a function like ?
- What is the general rule for determining the domain of a composite function?
- Can there be other restrictions on the domain besides the denominator being zero?
Tip: Always check for points where a function is undefined, such as divisions by zero or negative values under a square root, when finding the domain.
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Math Problem Analysis
Mathematical Concepts
Domain of a function
Composite functions
Algebra
Formulas
f(x) = 1/x
g(x) = 1/(5x)
(f ∘ g)(x) = f(g(x))
Theorems
The domain of a rational function is all real numbers except where the denominator equals zero.
Suitable Grade Level
Grades 10-12
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