Question

Let \(f\) be a function defined by \(f(x)=\sqrt{x}+\sin x\) on the interval \([0,2 \pi]\). a. Find an even function \(g\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(g(x)=f(x)\) for all \(x\) in \([0,2 \pi]\). b. Find an odd function \(h\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(h(x)=f(x)\) for all \(x\) in \([0,2 \pi]\).

Short answer

The even function extension of \(f(x)=\sqrt{x}+\sin x\) on the interval \([-2\pi, 2\pi]\) is given by: \(g(x)=\begin{cases} \sqrt{x}+\sin x &\text{for } x\in [0,2\pi]\\ \sqrt{-x}+\sin(-x) &\text{for } x\in [-2\pi,0) \end{cases}\) The odd function extension of \(f(x)=\sqrt{x}+\sin x\) on the interval \([-2\pi, 2\pi]\) is given by: \(h(x)=\begin{cases} \sqrt{x}+\sin x &\text{for } x\in [0,2\pi]\\ -\sqrt{-x}-\sin(-x) &\text{for } x\in [-2\pi,0) \end{cases}\)

Step-by-step solution

Use the properties of even functions

For an even function g, we know that \(g(-x)=g(x)\). So we need to find a function g(x) that satisfies this property and is equal to f(x) in the interval \([0,2\pi]\). In other words, we need to find a function g such that its value at -x equals its value at x.

Define g(x) based on f(x) and the properties of even functions

Since g(x) must maintain the properties of even functions and it must equal f(x) for all x in \([0,2\pi]\), we will define g(x) for positive x as \(g(x)=f(x)=\sqrt{x}+\sin x\). To define g(x) for negative values, let's use the property of even functions: \(g(-x)=g(x)=\sqrt{x}+\sin x\). Thus, \(g(x)=\sqrt{-x}+\sin(-x)\) for x in the interval \([-2\pi,0)\). We now have the even function g(x) defined on the interval \([-2\pi, 2\pi]\) such that \(g(x)=f(x)\) for all x in \([0,2\pi]\). #b. Finding the odd function h#

Use the properties of odd functions

For an odd function h, we know that \(h(-x)=-h(x)\). So we need to find a function h(x) that satisfies this property and is equal to f(x) in the interval \([0,2\pi]\). In other words, we need to find a function h such that its value at -x equals the negative of its value at x.

Define h(x) based on f(x) and the properties of odd functions

Since h(x) must maintain the properties of odd functions and it must equal f(x) for all x in \([0,2\pi]\), we will define h(x) for positive x as \(h(x)=f(x)=\sqrt{x}+\sin x\). To define h(x) for negative values, let's use the property of odd functions: \(h(-x)=-h(x)=-\sqrt{x}-\sin x\). Thus, \(h(x)=-\sqrt{-x}-\sin(-x)\) for x in the interval \([-2\pi,0)\). We now have the odd function h(x) defined on the interval \([-2\pi, 2\pi]\) such that \(h(x)=f(x)\) for all x in \([0,2\pi]\).

Key concepts

Symmetry in Functions

Symmetry in mathematics is a fundamental concept that can significantly simplify the study of functions. When a function exhibits symmetry, it means that its graph has a pattern which is mirrored across a certain line or point. This mirroring helps us understand the behavior of the function without having to compute every single value.

There are two primary types of symmetry relevant to functions: even and odd symmetry. An even function is symmetric with respect to the y-axis. Graphically, this means if you fold the graph along the y-axis, both halves would match. In simpler terms, for every point (x, y) on the graph, there is a corresponding point (-x, y). Mathematically, this is described by the equation \( g(-x) = g(x) \).

An odd function, on the other hand, is symmetric with respect to the origin. This means that rotating the graph 180 degrees around the origin would bring the graph onto itself. In effect, for every point (x, y) on the graph, there is a corresponding point (-x, -y). This relationship can be captured by the equation \( h(-x) = -h(x) \).

The concept of symmetry in functions not only helps in graphing but also in solving problems involving even and odd functions, as seen in the given textbook exercise.

Defining Functions with Intervals

Defining functions on specific intervals is a crucial skill in mathematics, particularly when dealing with real-world problems or when the function behaves differently in various domains. When we define functions with intervals, we are specifying the set of all possible input values, known as the domain, over which the function is intended to operate.

In the context of the exercise, the function \( f(x) = \sqrt{x} + \sin{x} \) is initially defined on the interval \( [0, 2\pi] \), which constrains x to be between 0 and \( 2\pi \), inclusive. However, creating an even function \( g \) and an odd function \( h \) based on \( f \) requires extending these definitions to include negative inputs as well. This extension is done over the interval \( [-2\pi, 2\pi] \).

To define a function with intervals correctly, it's vital to consider the properties of the function over the entire range. In the case of \( g(x) \) for negative inputs, the square root function \( \sqrt{x} \) does not have real values for negative x; therefore, this term is ignored when defining \( g \) for \( x < 0 \) as \( g(x) = \sin(-x) \), since the sine function is defined for all real numbers. This careful consideration of domain and properties ensures that the newly defined functions maintain mathematical consistency.

Properties of Even and Odd Functions

Understanding the properties of even and odd functions is essential for solving many mathematical problems, including those involving function transformations and symmetry. For example, when dealing with integrals or series, knowing the parity of a function can simplify the work.

For even functions, one of the key properties is that \( g(x) = g(-x) \) for all x within the function’s domain. This characteristic allows for simplifications in many situations, such as evaluating integrals over symmetric intervals since the area under the curve from \( -a \) to 0 is the same as from 0 to \( a \).

Odd functions have the complementary property that \( h(-x) = -h(x) \). This leads to the result that the integral of an odd function over a symmetric interval around the origin is always zero, as the positive and negative areas cancel each other out. Also, the product of two even functions or two odd functions is an even function, while the product of an even function and an odd function is odd.

These properties of even and odd functions not only make abstract concepts more tangible but also provide an arsenal of tools for tackling higher-level mathematics. They are exemplified in the step-by-step solution to the given exercise, which requires a thoughtful application of these properties to define the even function \( g \) and the odd function \( h \) correctly over the extended intervals.