Chapter 2: Problem 31
Decide whether the function is even, odd, or neither. \(f(x)=x \sqrt{4-x^{2}}\)
Short Answer
Expert verified
The function \(f(x)=x \sqrt{4-x^{2}}\) is odd.
Step by step solution
01
Identify the Given Function
The given function is \(f(x)=x \sqrt{4-x^{2}}\).
02
Check if the function is Even
To check if \(f(x)\) is an even function, we insert \(-x\) into the equation instead of \(x\), if the outcome is equal to \(f(x)\) then it is an even function. We have \(f(-x)=(-x) \sqrt{4-(-x)^{2}}=(-x) \sqrt{4-x^{2}}\) which is not equal to \(f(x)\). Therefore, the function is not even.
03
Check if the function is Odd
To check if \(f(x)\) is an odd function, we insert \(-x\) into the equation instead of \(x\), if the outcome is equal to \(-f(x)\) then it is an odd function. However, since we found that inserting \(-x\) into the equation gives \(-f(x)\), so we can say that the function is odd.
04
Conclusion
After evaluating the function, we conclude that \(f(x)=x \sqrt{4-x^{2}}\) is an odd function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetry of Functions
Understanding the symmetry of functions is crucial when studying mathematics, as it helps in graphing and analyzing the behavior of functions. The symmetry of a function refers to its invariance under certain operations, typically reflections across an axis.
For a function to be even, its graph must be symmetric about the y-axis. This means that for every point (x, y) on the graph, there is a corresponding point (-x, y). Algebraically, a function is even if for every input value x, the equality f(x) = f(-x) holds true.
On the other hand, an odd function shows symmetry about the origin. In graphical terms, for every point (x, y), there is a corresponding point (-x, -y) that also lies on the graph. Algebraically, a function is odd if f(-x) = -f(x) for all x in the domain. There is also a possibility for a function to be neither even nor odd if it doesn't display either symmetry.
For instance, in the exercise involving the function
\(f(x)=x \sqrt{4-x^{2}}\),
it was determined that the function is odd by checking if the output for \(-x\) yields the opposite value \(-f(x)\). Understanding this concept assists students in predicting the behavior of functions and enhances their graphing skills.
For a function to be even, its graph must be symmetric about the y-axis. This means that for every point (x, y) on the graph, there is a corresponding point (-x, y). Algebraically, a function is even if for every input value x, the equality f(x) = f(-x) holds true.
On the other hand, an odd function shows symmetry about the origin. In graphical terms, for every point (x, y), there is a corresponding point (-x, -y) that also lies on the graph. Algebraically, a function is odd if f(-x) = -f(x) for all x in the domain. There is also a possibility for a function to be neither even nor odd if it doesn't display either symmetry.
For instance, in the exercise involving the function
\(f(x)=x \sqrt{4-x^{2}}\),
it was determined that the function is odd by checking if the output for \(-x\) yields the opposite value \(-f(x)\). Understanding this concept assists students in predicting the behavior of functions and enhances their graphing skills.
Algebraic Functions
Algebraic functions represent one of the pillars of algebra and are defined by using algebraic expressions that involve a finite combination of the basic operations: addition, subtraction, multiplication, division, and extraction of roots. These functions can be expressed by equations that relate two variables.
Examples of algebraic functions include polynomial functions, rational functions, and radical functions. The function presented in the given exercise \(f(x)=x \sqrt{4-x^{2}}\)is classified as a radical function because it includes a square root, a type of root extraction. This function is particularly interesting as it not only showcases the operation of multiplication but also the interaction between a linear term \(x\)and a square root of a quadratic expression. It's vital for students to recognize various algebraic functions to navigate through different problem-solving scenarios with ease and develop a robust understanding of function behavior and transformations.
Examples of algebraic functions include polynomial functions, rational functions, and radical functions. The function presented in the given exercise \(f(x)=x \sqrt{4-x^{2}}\)is classified as a radical function because it includes a square root, a type of root extraction. This function is particularly interesting as it not only showcases the operation of multiplication but also the interaction between a linear term \(x\)and a square root of a quadratic expression. It's vital for students to recognize various algebraic functions to navigate through different problem-solving scenarios with ease and develop a robust understanding of function behavior and transformations.
Function Transformation
Function transformation involves changing a function's graph in some way, which may include shifting, stretching, compressing, or reflecting. These transformations can be applied to the basic function's graph to effectively alter the shape or position without changing its core properties.
Common transformations include:
Take the function \(f(x)=x \sqrt{4-x^{2}}\).If we wanted to reflect it across the y-axis (a transformation), we'd replace x with -x. However, since this function is already odd, reflecting it in this manner will result in the same function, as odd functions are inherently symmetric about the origin. These transformations form an integral part of analyzing and understanding various attributes of different functions.
Common transformations include:
- Vertical shifts, produced by adding or subtracting a constant.
- Horizontal shifts, resulting from adding or subtracting a number inside the function's argument.
- Vertical stretches and compressions, achieved by multiplying the function by a constant.
- Horizontal stretches and compressions, performed by multiplying the input variable by a constant.
- Reflections, similar to determining if a function is even or odd as explored in the exercise.
Take the function \(f(x)=x \sqrt{4-x^{2}}\).If we wanted to reflect it across the y-axis (a transformation), we'd replace x with -x. However, since this function is already odd, reflecting it in this manner will result in the same function, as odd functions are inherently symmetric about the origin. These transformations form an integral part of analyzing and understanding various attributes of different functions.
