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Chapter 3: Problem 78

Determine whether each function is odd, even, or neither. \(f(x)=x \cos x\)

Short Answer

Expert verified
The function is odd.

Step by step solution

01

Determine the function properties

Determine whether the given function is odd, even, or neither by analyzing its symmetry. The given function is \( f(x) = x \cos x \).
02

Calculate \( f(-x) \)

Substitute \( -x \) into the function: \[ f(-x) = (-x) \cos(-x) \].
03

Simplify \( f(-x) \) using trigonometric identity

Use the trigonometric identity \( \cos(-x) = \cos(x) \): \[ f(-x) = -x \cos(x) \].
04

Compare \( f(x) \) and \( f(-x) \)

Compare the expressions for \( f(x) \) and \( f(-x) \): \[ f(x) = x \cos x \]\[ f(-x) = -x \cos x \].
05

Determine whether the function is odd, even, or neither

Since \( f(-x) = -f(x) \), the function \( f(x) = x \cos x \) is odd.

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Key Concepts

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

Function Symmetry
When dealing with functions, one important aspect to consider is their symmetry. Symmetry gives us insights about the function's behavior without having to graph it. There are two main types of symmetry: even and odd. A function is considered **even** if it satisfies the condition: ewline ewline

ewline where the graph is symmetric with respect to the y-axis.ewline ewline

Similarly, a function is considered **odd** if it satisfies: ewline ewline

If a function doesn't fall into either category, it is considered **neither even nor odd**.ewline
Trigonometric Identity
In trigonometry, identities are equations that are true for all values of the variables within a certain range. One useful identity for analyzing function symmetry is the cosine function: ewline ewline

Cosine is an **even function**: it remains the same if you replace its variable with its negative. This property is particularly useful when simplifying expressions: ewline ewline

For example, if we have cos(-x), it simplifies to cos(x). Recognizing this identity helps in determining whether a composite function involving cosine is even or odd.
Odd Function
An odd function has a distinctive property where the function鈥檚 value at opposite inputs are negatives of each other. This can be expressed as:

Which means if you take any x in the function and replace it with -x, and then if you get the negative of the original function, it tells you that the function is odd. Let's take the function from the exercise: ewline Substituting -x, we get for

f(-x) = (-x) cos(-x) . Simplifying using the trig identity we get = -x cos(x). This confirms f(x) = x cos(x) is odd since f(-x) = -f(x). Understanding odd functions is helpful to identify reflectional symmetry across the origin.

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Most popular questions from this chapter

Simplify each expression by applying the odd/even identities, cofunction identities, and cosine of a sum or difference identities. Do not use a calculator: $$ \cos \left(4^{\circ}\right) \cos \left(9^{\circ}\right)+\cos \left(86^{\circ}\right) \cos \left(81^{\circ}\right) $$

Let \(f(x)=\sin (x), g(x)=x+2,\) and \(h(x)=3 x\). Find \(g(f(h(x)))\) and \(h(g(f(x)))\)

Find the exact value of \(\cos (\alpha-\beta)\) if \(\sin \alpha=-4 / 5\) and \(\cos \beta=12 / 13,\) with \(\alpha\) in quadrant III and \(\beta\) in quadrant IV.

Use identities to simplify each expression. Do not use a calculator. \(\frac{\sin 12^{\circ}}{1+\cos 12^{\circ}}\)

Verify that each equation is an identity. \(\tan (\pi / 4+x)=\cot (\pi / 4-x)\)
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