Exploring the Characteristics of a Squared-Reciprocal Function | A Mathematical Analysis

Squared-Reciprocal Function

A squared-reciprocal function, also known as a reciprocal square function, is a mathematical function of the form f(x) = 1/x^2

A squared-reciprocal function, also known as a reciprocal square function, is a mathematical function of the form f(x) = 1/x^2. It is a combination of two operations: squaring and taking the reciprocal.

Let’s break down the function and understand its characteristics:

1. Reciprocal: The reciprocal of a number x is 1/x. In a squared-reciprocal function, we take the reciprocal of the input value x.

2. Squaring: The square of a number x is x^2. In a squared-reciprocal function, we take the reciprocal of x and then square it.

Putting it all together, for any given input value x, the squared-reciprocal function produces the output value of 1/x^2.

Some important characteristics of a squared-reciprocal function:

1. Domain: The function is defined for all real numbers except x = 0, as taking the reciprocal of 0 is undefined.

2. Range: The function has a range of (0, ∞), meaning that the output values are always positive and infinitesimally close to 0 as x approaches positive or negative infinity.

3. Symmetry: The squared-reciprocal function is symmetric about the y-axis. This means that if we find the value of f(x) for a particular x, we will get the same value if we substitute -x as the input.

4. Asymptotes: The function has two vertical asymptotes. One is located at x = 0, where the function approaches positive infinity as x approaches 0 from the left side, and negative infinity as x approaches 0 from the right side. The other asymptote is at the y-axis, as there is no value of x for which f(x) can be 0.

5. Decreasing behavior: The squared-reciprocal function is always decreasing. As x increases or decreases, the value of 1/x^2 decreases, getting closer to 0.

In summary, a squared-reciprocal function is a mathematical function that takes the reciprocal of the input value, and then squares it to create the output value. It has a number of unique characteristics, including a defined domain and range, symmetry, asymptotes, and decreasing behavior.

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