Odd Function
An odd function is a type of mathematical function where the function’s value changes sign when the input is negated
An odd function is a type of mathematical function where the function’s value changes sign when the input is negated. In other words, if the function is denoted as f(x), then it satisfies the property:
f(-x) = -f(x) for all x in the domain of the function.
One common example of an odd function is the function f(x) = x. Let’s evaluate this function for some values of x:
f(1) = 1
f(-1) = -(-1) = 1
f(2) = 2
f(-2) = -(-2) = 2
As you can see, when we negate the input, the function’s value also negates. This property holds true for all values of x, making it an odd function.
Another example of an odd function is f(x) = x^3. Let’s evaluate this function as well:
f(1) = 1
f(-1) = -(-1)^3 = -(-1) = -1
f(2) = 2^3 = 8
f(-2) = -(-2)^3 = -(-8) = -8
Again, we see that the function’s value changes sign when the input is negated. This is a characteristic property of odd functions.
One important property of odd functions is that their graphs exhibit symmetry about the origin (0,0). This means that if you fold the graph in half along the y-axis, the resulting halves will be mirror images of each other.
Odd functions can be visualized as having rotational symmetry of 180 degrees about the origin. This means that if you rotate the graph of an odd function by 180 degrees (half a complete rotation) about the origin, it will retain the same shape.
Odd functions have various applications in mathematics and physics. They are often used to model quantities that exhibit a certain kind of symmetry, such as electrical signals and waveforms.
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