Math Problem Statement

what type of function is y=1/x^2+1

Solution

The function y=1x2+1y = \frac{1}{x^2 + 1} is a rational function, because it is the ratio of two polynomials. Specifically, it's the ratio of the constant 11 over the quadratic expression x2+1x^2 + 1.

Characteristics of this function:

  1. Domain: The function is defined for all real numbers xx, because the denominator x2+1x^2 + 1 is always positive and never equals zero (there are no real roots for x2+1=0x^2 + 1 = 0).

  2. Range: The output (or range) of the function is bounded between 0<y10 < y \leq 1. The value of yy is always positive, but never zero, and the largest value occurs at x=0x = 0, where y=1y = 1.

  3. Asymptotes:

    • Horizontal asymptote: As xx \to \infty or xx \to -\infty, the value of the function approaches y=0y = 0, so there is a horizontal asymptote at y=0y = 0.
    • Vertical asymptote: There are no vertical asymptotes, since the denominator x2+1x^2 + 1 never equals zero.
  4. Even function: The function is symmetric about the y-axis because f(x)=f(x)f(x) = f(-x). This means it is an even function.

  5. Decreasing behavior: The function is strictly decreasing for x>0x > 0 and for x<0x < 0, meaning it becomes smaller as x|x| increases.

Would you like further details on how to derive any of these characteristics or ask any specific questions about the function?


Here are five related questions:

  1. What are the properties of rational functions in general?
  2. How do you determine the horizontal asymptote of a rational function?
  3. Can you find the first derivative of y=1x2+1y = \frac{1}{x^2 + 1} to determine the rate of change?
  4. How do we graph rational functions with asymptotes?
  5. How do you solve rational inequalities involving similar functions?

Tip: When analyzing rational functions, always check the domain first by ensuring the denominator does not equal zero!

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Algebra
Asymptotes
Symmetry
Domain and Range

Formulas

y = 1 / (x^2 + 1)

Theorems

Properties of Rational Functions
Horizontal Asymptotes
Even Functions

Suitable Grade Level

Grades 10-12