Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false. \text { The function } y=\sin ^{2} x \text { is an odd function. }

Short answer

The statement is false. The product of two odd functions, \(f(x)\) and \(g(x)\), results in an even function, \(h(x)=f(x)g(x)\), because \(h(-x)=f(-x)g(-x)=(-f(x))(-g(x))=f(x)g(x)=h(x)\), which satisfies the definition of an even function.

Step-by-step solution

Recall the definition of odd and even functions

An even function is a function for which \(f(-x) = f(x)\) for all \(x\) in its domain. An odd function is a function for which \(f(-x) = -f(x)\) for all \(x\) in its domain.

Determine the product of two odd functions

We need to determine whether the product of two odd functions, say \(f(x)\) and \(g(x)\), is an odd function. Let's call this product function as \(h(x) = f(x)g(x)\). Now let's compute \(h(-x)\).

Compute h(-x)

Let's find the value of \(h(-x)\): \(h(-x) = f(-x)g(-x)\)

Apply the definition of odd functions

Since both \(f\) and \(g\) are odd functions, we can use the definition of odd functions (\(f(-x) = -f(x)\) and \(g(-x) = -g(x)\)) to replace \(f(-x)\) and \(g(-x)\) in the expression for \(h(-x)\): \(h(-x) = (-f(x))(-g(x))\)

Simplify product of two odd functions

Now, we will simplify the product of \(-f(x)\) and \(-g(x)\) in the expression for \(h(-x)\): \(h(-x) = (-1)(-1)f(x)g(x) = f(x)g(x) = h(x)\)

Determine if the statement is true or false

By Step 5, we find out that \(h(-x) = h(x)\), which is the definition of an even function. Therefore, the statement is false. The product of two odd functions is an even function, not an odd function.

Key concepts

Understanding \(sin^2(x)\)

When studying trigonometric functions, \(sin^2(x)\) often appears in equations and can cause confusion due to its notation. The notation \(sin^2(x)\) simply means the sine of \(x\) squared, or \( (sin(x)) ^2 \).

It's important to recognize that squaring the sine of an angle doesn't affect the periodic nature of the function, but it does impact its symmetry with respect to the y-axis. While the sine function itself is an odd function, once squared, it behaves differently than its original odd function in terms of symmetry, which brings us to the need to reassess its properties within the context of odd and even function definitions.

Product of Two Odd Functions

To understand the product of two odd functions, let's first grasp what an odd function is. An odd function is symmetrical with respect to the origin, meaning that flipping it across both axes retains its shape. Mathematically, for a function \(f\), being odd means \(f(-x) = -f(x)\) for all \(x\) in the function's domain.

When we multiply two odd functions \(f(x)\) and \(g(x)\) together, as seen in the step-by-step solution, we find that the resulting function \(h(x) = f(x)g(x)\) fulfills the criteria of an even function, not an odd one. This happens because the negatives from both \(f(-x)\) and \(g(-x)\) cancel each other out, leaving an unchanged function \(h(x)\) when \(x\) is replaced with \( -x \). In simpler terms, multiplying two odd functions results in an even function due to the negation of oddness.

Even Function Definition

An even function has a particular symmetry that makes it a mirror image across the y-axis. This defining characteristic can be captured mathematically: for any even function \(f\), \(f(-x) = f(x)\) for every \(x\) in its domain. Some well-known examples of even functions include \(x^2\), \(cos(x)\), and our product of two odd functions from the example given earlier.

For a visual learner, imagine folding a graph along the y-axis. If the two halves align perfectly, the function graphed is even. This concept is critical in understanding the symmetries in functions and plays a role in various areas of mathematics, including integral calculus, where the properties of even functions simplify the evaluation of integrals.

Odd Function Definition

Conversely, odd functions are characterized by their rotational symmetry around the origin (rotated 180 degrees). The algebraic test for oddness is that for every \(x\) in the domain of the function \(f\), \(f(-x) = -f(x)\) holds true. The classic example of an odd function is \(f(x) = x^3\), but the sine function, \(sin(x)\), is also odd.

An odd function's graph exhibits a sort of balance, such that if a point \( (a, b) \) is on the function, then \( (-a, -b) \) is also on it. This property is essential in understanding the behavior of functions under transformations and has implications in series expansions, particularly Fourier series, where functions are decomposed into sums of sines and cosines.