Math Problem Statement
Solution
It looks like you've shared a mathematical problem asking for the values of for which various functions are continuous.
Let's go over the functions one by one:
a)
This is a polynomial function, which is continuous for all real numbers. There are no restrictions.
b)
This is another polynomial, and like the previous one, it is continuous for all real numbers.
c)
This is a rational function. For a rational function to be continuous, the denominator cannot be zero. Therefore, we must find the values of that make the denominator zero and exclude those from the domain.
Denominator: , so solving for , we get:
Thus, or . These values will make the denominator zero, so the function is not continuous at and . The function is continuous for all .
d)
This is a square root function. For square roots to be defined, the expression inside the square root must be non-negative. Therefore: This function is continuous for all .
e)
This is an exponential function. Exponential functions are continuous for all real numbers, so this function is continuous for all .
f)
This is another rational function. The denominator cannot be zero, so we must ensure . Since is always positive (no real value of will make this expression zero), the function is continuous for all real numbers.
Would you like more details on any of the steps, or is there something else you need clarification on?
Here are some related questions to deepen your understanding:
- What are the general rules for determining continuity in rational functions?
- How do you find the domain of a square root function?
- How can you analyze the continuity of a function involving an absolute value?
- What are the main types of discontinuities for rational functions?
- How do you deal with piecewise functions when determining continuity?
Tip: Always remember that rational functions are continuous except where their denominators equal zero!
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Math Problem Analysis
Mathematical Concepts
Polynomials
Rational Functions
Square Roots
Exponential Functions
Formulas
Denominator cannot be zero for rational functions
Square root requires non-negative expressions
Theorems
Continuity of polynomials
Continuity of rational functions
Continuity of square root functions
Continuity of exponential functions
Suitable Grade Level
Grades 9-12
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