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Magazine Chapter 4: Problem 10
Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\sqrt{3+2 x-x^{2}} ;(-1,3),[-1,3],[-1,3),(-1,3] $$
Short Answer
Expert verified
The function is continuous on all given intervals.
Step by step solution
01
Understand the Function
Consider the function given: \( f(x)=\sqrt{3+2x-x^2} \). This is a square root function, which is defined as long as the expression under the square root is non-negative, i.e., \( 3+2x-x^2 \geq 0 \).
02
Determine the Domain
To find the domain, solve the inequality: \( 3+2x-x^2 \geq 0 \). Rewrite it in standard form: \( -x^2+2x+3 \geq 0 \). This is a quadratic inequality.
03
Solve the Quadratic Inequality
Find the roots of the quadratic equation \( -x^2+2x+3 = 0 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) where \( a = -1 \), \( b = 2 \), and \( c = 3 \), we get roots: \( x = -1 \) and \( x = 3 \). The inequality \( -x^2+2x+3 \geq 0 \) holds between these two roots.
04
Analyze the Continuity
The function \( f(x) \) is continuous wherever it is defined. It turns out that \( -1 \leq x \leq 3 \) is the interval where the expression under the square root is non-negative. We now evaluate the function's behavior at the boundaries of given intervals.
05
Evaluate (-1, 3)
For \( (-1, 3) \), the function is continuous since it is within the defined domain and doesn't include the endpoints.
06
Evaluate [-1, 3]
For \( [-1, 3] \), the function is continuous because it includes both endpoints, within which the function is also defined.
07
Evaluate [-1, 3)
For \( [-1, 3) \), the function is continuous since we include \( -1 \), but not \( 3 \), and both fall within the domain.
08
Evaluate (-1, 3]
For \( (-1, 3] \), the function is continuous since we include \( 3 \), but not \( -1 \), and both are within the domain.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Inequality
A quadratic inequality involves a quadratic expression set within an inequality, typically in the form of \(ax^2 + bx + c \geq 0\) or \(ax^2 + bx + c \leq 0\). It is crucial to determine where the quadratic expression is non-negative or non-positive. To solve a quadratic inequality:
- First, solve the corresponding quadratic equation \(ax^2 + bx + c = 0\) to find its roots.
- For example, in the function \(f(x)=\sqrt{3+2x-x^2}\), we considered \(3 + 2x - x^2 \geq 0\).
- Rewrite it as \(-x^2 + 2x + 3 \geq 0\). The roots found using the quadratic formula, \(x = -1\) and \(x = 3\), divide the number line into intervals.
- Analyze each interval to determine where the inequality holds true. For the given function, the inequality \(-x^2 + 2x + 3 \geq 0\) holds between the roots \(-1\) and \(3\), meaning within the interval \([-1, 3]\). This interval is crucial in understanding where the square root function is defined.
Function Domain
The domain of a function comprises all the input values (x-values) for which the function is defined. For a square root function, the expression inside the square root must be non-negative.
In the given function \(f(x)=\sqrt{3+2x-x^2}\), we determine the domain by ensuring:
In the given function \(f(x)=\sqrt{3+2x-x^2}\), we determine the domain by ensuring:
- \(3 + 2x - x^2 \geq 0\) must hold true.
- Earlier, we determined this inequality holds for \(-1 \leq x \leq 3\). Therefore, the domain of \(f(x)\) is \([-1, 3]\). Understanding the domain helps in knowing where the function is valid for given values of x.
Square Root Function
Square root functions contain a square root operation, primarily affecting their definition and range. For them to be real-valued, the expression under the square root must be non-negative.
In \(f(x)=\sqrt{3+2x-x^2}\):
In \(f(x)=\sqrt{3+2x-x^2}\):
- We need \(3+2x-x^2 \geq 0\) for the function to have real outputs.
- Through continuity analysis, we know that within the domain \([-1, 3]\), the square root expression stays non-negative.
- Beyond this interval, the function \(f(x)\) becomes undefined. Understanding a square root function means ensuring the expression inside remains valid for real numbers across required intervals.
Interval Notation
Interval notation is a compact way to describe a set of numbers along a continuum. It specifies the start and end points and whether these points are included or not, using parentheses \(( )\) for exclusion and brackets \([ ]\) for inclusion. For the function \(f(x)=\sqrt{3+2x-x^2}\):
- Knowing the domain \([-1, 3]\) helps us describe where the function behaves continuously.
- Open interval \((-1, 3)\) tells us to exclude -1 and 3, meaning our function doesn't include these endpoints.
- Closed interval \([-1, 3]\) includes both ends, showing the continuous nature from, and including, -1 to 3.
- Mixed intervals such as \([-1, 3)\) and \((-1, 3]\) respectively show the inclusion of -1 or 3, but not both, adding to understanding where the function keeps continuity. Interval notation provides a clear and concise way to represent domains and ranges, crucial for analyzing function behavior.
