This is the most important functions. The rule it describes is as follows: use the x value as the y value. The x and the y value in each ordered pair are identically the same, hence, it is called the identity function.
It is linear. It is a first degree polynomial because it may be written as y = x or where n the exponent is 1 and a the coefficient is 1.
Its slope is 1 because the coefficient of the x term is 1. Its reciprocal is 1/x. It is its own inverse: the inverse function for y = x is y = x.
The identity function may be thought of as the basis for all other functions. The linear function y = x + 3, or just x + 3, may be thought of as the sum of the identity function, x, and the constant function 3. The linear function y = 4x, or just 4x, may be thought of as the product of the identity function, x, and the constant function 4. The quadratic function y = x², or just x², may be thought of as the square of the identity function, x · x, or xx, or x².
The opposite function returns the opposite of whatever number is operated upon. If one takes the opposite of a positive, the result is negative. If one takes the opposite of a negative, the result is positive. The opposite of zero is zero.
Experiment with this yourself. Enter a number here: . (Enter negatives as "-x" rather than "- x.") Press: . The oposite of your number is: .
It is linear. Its slope is -1. Its reciprocal is -1/x. It is its own inverse: the inverse function is -x.
Graphically the opposite is useful in reflecting a curve about a vertical axis or a horizontal axis.
The slope of the reciprocal function is -1/x². It is its own inverse: the inverse of the reciprocal function is the reciprocal function.
It is a hyperbola, asymptotic to both the x- and y- axes. The reciprocal plays an important role in the creation of other functions. When a function increases, its reciprocal function decreases. When a function achieves a minimum, its reciprocal achieves a maximum. The reciprocal of the sine function is called the cosecant. A reciprocal function is a factor of each rational function.
This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).
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