Math Problem Statement
Find the domain and the range of the following real functions: (a) π(π₯) = (π₯2 β 4)/(x+2)
(f) π(π₯)=|π₯|βπ₯ /(2π₯)
(g)π(π₯)= π₯β3/( 2π₯ + 1)
(h) π(π₯) = 1 /(1βπ₯2)
(i) π(π₯) = βπ₯2 β 4
(j) π(π₯)= π₯+2/( |π₯ + 2|)
Solution
Let's analyze each function step by step to find the domain (the set of all possible input values) and the range (the set of all possible output values).
(a)
Domain:
- The denominator must not be zero, so .
- Thus, the domain is:
Simplification:
- The numerator can be factored as , so the function simplifies: For , this simplifies to:
Range:
- Since for all , the range is all real numbers:
(f)
Domain:
- The denominator must not be zero, so .
- Thus, the domain is:
Simplification:
- For , , so .
- For , , so .
Range:
- For , .
- For , .
- Therefore, the range is:
(g)
Domain:
- The denominator , so .
- Thus, the domain is:
Range:
- ( f(x
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Math Problem Analysis
Mathematical Concepts
Functions
Domain and Range
Absolute Value
Square Roots
Rational Functions
Formulas
f(x) = (x^2 - 4)/(x + 2)
f(x) = |x| β x / (2x)
f(x) = (x - 3)/(2x + 1)
f(x) = 1/(1 - x^2)
f(x) = β(x^2 - 4)
f(x) = (x + 2)/|x + 2|
Theorems
Domain exclusion for denominators
Properties of absolute value
Range determination for rational functions
Suitable Grade Level
Grades 10-12
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