Problem 2 Are the following functions even... [FREE SOLUTION]

Chapter 11: Problem 2

Are the following functions even, odd, or neither even nor add? $$\sin \left(x^{2}\right), \sin ^{2} x, x \sinh x,\left|x^{2}\right|, e^{-\infty}, x e^{x}, \tan 2 x, x /\left(1+x^{2}\right)$$

Short Answer

Expert verified

$ \sin(x^2), \sin^2(x), |x^2|, e^{- ext{Infinity}} $ are even; $ x e^x $ is neither; $ x \sinh(x), \tan(2x), \frac{x}{1+x^2} $ are odd.

Step by step solution

Understanding Even and Odd Functions

A function is even if $ f(x) = f(-x) $ for all $ x $. It is odd if $ f(-x) = -f(x) $ for all $ x $. If neither condition is satisfied, the function is neither even nor odd.

Analyzing $ \sin(x^2) $

For even function characteristics, check $ \sin((-x)^2) = \sin(x^2) $. Since $ x^2 = (-x)^2 $, $ \sin(x^2) $ is even.

Analyzing $ \sin^2(x) $

$ \sin^2(x) $ can be rewritten as $ [\sin(x)]^2 $. As $ (-\sin(x))^2 = \sin^2(x) $, the function is even.

Analyzing $ x \sinh(x) $

$ \sinh(x) $ is an odd function. Therefore, $ x \sinh(x) $ gives $(-x) \sinh(-x) = x \sinh(x) $, making it even because it matches the same logic as even functions.

Analyzing $ |x^2| $

$ |x^2| = x^2 $, which is an even function because $ x^2 = (-x)^2 $. Thus, $ |x^2| $ is also even.

Evaluating $ e^{- ext{Infinity}} $

$ e^{- ext{Infinity}} $ evaluates to $ f(x) = 0 $. Constants are even functions, since $ f(x) = f(-x) $ is valid.

Analyzing $ x e^x $

For $ x e^x $, $ f(-x) = -x e^{-x} $ does not equal $ f(x) $ or $-f(x)$. Thus, $ x e^x $ is neither even nor odd.

Analyzing $ \tan(2x) $

$ \tan(2x) $ is an odd function, since $ \tan(-2x) = -\tan(2x) $.

Analyzing $ \frac{x}{1+x^2} $

For $ \frac{x}{1+x^2} $, compute $ f(-x) = \frac{-x}{1+(-x)^2} = -\frac{x}{1+x^2} $, meaning it's an odd function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Properties

Understanding the properties of functions plays a crucial role in analyzing whether a function is even, odd, or neither. The properties give essential clues:

**Even Function:** A function is even if it holds the property that $ f(x) = f(-x) $ for all x. Graphically, these functions exhibit symmetry about the y-axis, meaning the graph looks the same on both sides of this axis.
**Odd Function:** A function is odd if $ f(-x) = -f(x) $ for all x. These functions show symmetry around the origin; which means if it includes a point $(a, b)$, it also includes $(-a, -b)$.
**Neither Even nor Odd:** If a function doesn鈥檛 satisfy even or odd properties, it falls into this category.

Knowing these properties helps not only in understanding the function types based on symmetry but also boosts solving equations and integrals efficiently.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are foundational components in mathematics.

**Sine and Cosine:** The sine function, $ \sin(x) $, is odd because $ \sin(-x) = -\sin(x) $. On the other hand, cosine, $ \cos(x) $, is even; $ \cos(-x) = \cos(x) $.
**Tangent:** The tangent function, $ \tan(x) $, is an odd function. When applied through transformation as in $ \tan(2x) $, it retains its oddness because $ \tan(-2x) = -\tan(2x) $.

Moreover, when you analyze transformations, like squaring terms or multiplying them, those are critical in determining the even or odd nature of the resulting function. Take $ \sin(x^2) $, here, the inside term $ x^2 $ is even, leading the entire function to be even. Similarly, $ \sin^2(x) = (\sin(x))^2 $ is even since it results in positive outputs irrespective of the sign of x.

Hyperbolic Functions

Hyperbolic functions are analogs to trigonometric functions and have similar but distinct properties. Common hyperbolic functions include $ \sinh(x) $ and $ \cosh(x) $.

**Hyperbolic Sine:** The $ \sinh(x) $ function behaves like the sine function in that it's odd; $ \sinh(-x) = -\sinh(x) $.
**Hyperbolic Cosine:** The $ \cosh(x) $ function mimics the cosine function and is even, satisfying $ \cosh(-x) = \cosh(x) $.

Understanding these properties helps, especially when they combine with polynomials or other variables, to determine overall symmetry.
For instance, examining $ x \sinh(x) $ results in an even function. Despite $ \sinh(x) $ being odd, the product with $ x $ (also odd) leads to $ x \sinh(x) $ following an even property overall.

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91影视

Are the following functions even, odd, or neither even nor add? $$\sin \left(x^{2}\right), \sin ^{2} x, x \sinh x,\left|x^{2}\right|, e^{-\infty}, x e^{x}, \tan 2 x, x /\left(1+x^{2}\right)$$

Short Answer

Step by step solution

Understanding Even and Odd Functions

Analyzing \( \sin(x^2) \)

Analyzing \( \sin^2(x) \)

Analyzing \( x \sinh(x) \)

Analyzing \( |x^2| \)

Evaluating \( e^{- ext{Infinity}} \)

Analyzing \( x e^x \)

Analyzing \( \tan(2x) \)

Analyzing \( \frac{x}{1+x^2} \)

Key Concepts

Function Properties

Trigonometric Functions

Hyperbolic Functions

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