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SELFMADE
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What criteria decides whether a trig function is an odd or even?
The same criterion as any other odd function: f(-x) = -f(x).SELFMADE said:What criteria decides whether a trig function is an odd or even?
Mark44 said:Graphically speaking, an even function is its own reflection across the y-axis, which makes f(-x) = f(x). An odd function is its own reflection around the origin. This type of reflection is equivalent to a reflection across the x-axis, and then a reflection across the y-axis (or vice versa). This means that if you take, for example, the graph of y = tan x for x > 0, and reflect it across the x-axis, and then the y-axis, it will superimpose exactly on the the half of the graph of y = tan x for x < 0.
SELFMADE said:What I don't understand is
What is a "reflection"?
What is the difference between reflection across an axis and reflection around the origin?
What it means to reflect the graph?
An odd function is a mathematical function that satisfies the condition f(-x) = -f(x) for all values of x. This means that when the input of the function is multiplied by -1, the output of the function is also multiplied by -1. Graphically, an odd function will have symmetry about the origin.
When a function is considered odd, it means that it satisfies the condition f(-x) = -f(x) for all values of x. This property is important in mathematics because it allows for simplification and symmetry in equations and graphs.
To determine if a function is odd, you can use the symmetry test f(-x) = -f(x). If the function satisfies this condition for all values of x, then it is considered odd. Another way to check is by graphing the function and looking for symmetry about the origin.
Odd functions are important in math because they allow for simplification and symmetry in equations and graphs. They also have specific properties, such as having a zero at the origin and being antisymmetric, which can be useful in problem-solving and analysis.
Odd functions are used in real life in various fields such as physics, engineering, and economics. For example, in physics, odd functions are used to describe the relationship between force and displacement in simple harmonic motion. In economics, odd functions are used to model cost functions and profit functions. They are also used in signal processing and digital image processing to separate noise from a signal.