Math Problem Statement
f (x) = 5 √x − 3 (c) f (x) = 8x + 1 x2 + 3x + 2 (d) f (x) = 8x + 1 √x2 + 3x + 2 1
Solution
It seems like you have multiple function expressions here that need to be analyzed or simplified. Let’s go through each one individually to clarify and address them properly.
1. Function
This function represents a square root function with a linear transformation:
- The term suggests the function is defined for .
- The transformation scales the output by 5.
- Subtracting 3 shifts the function downward by 3 units.
2. Function
This is a rational function where:
- The numerator is .
- The denominator is .
- The domain is restricted where the denominator is not zero. To find where it’s undefined, set , which factors as . So, and are not in the domain.
3. Function
This function is similar to the previous one but has a square root in the denominator:
- The square root restricts the function further since must be non-negative and cannot be zero.
- Since the expression under the square root factors as , the domain restrictions are except for and .
Summary of Important Points:
- The first function is defined for .
- The second function is undefined at and .
- The third function is defined for but not at and .
Would you like me to dive deeper into any of these functions or explore specific properties like asymptotes, intercepts, or graphs?
Related Questions:
- How do you determine the domain of a function with a square root?
- What are the asymptotes of the function ?
- How do you factor quadratic expressions like ?
- What is the behavior of as approaches infinity?
- How do you simplify expressions involving square roots in the denominator?
Tip:
When analyzing the domain of a function, always look out for restrictions such as divisions by zero and square roots of negative numbers.
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Math Problem Analysis
Mathematical Concepts
Algebra
Square Root Functions
Rational Functions
Quadratic Equations
Formulas
f(x) = 5√x − 3
f(x) = (8x + 1) / (x^2 + 3x + 2)
f(x) = (8x + 1) / √(x^2 + 3x + 2)
Theorems
Domain of functions with square roots: x ≥ 0
Domain of rational functions: denominator cannot be zero
Quadratic factorization: x^2 + 3x + 2 = (x + 1)(x + 2)
Suitable Grade Level
Grades 10-12