Math Problem Statement

find the domain and range of each function 1+x^2

Solution

To find the domain and range of the function f(x)=1+x2f(x) = 1 + x^2, let's analyze its properties.

Domain:

The domain of a function consists of all possible values of xx for which the function is defined.

  • The given function f(x)=1+x2f(x) = 1 + x^2 is a polynomial function.
  • Polynomial functions are defined for all real numbers.

Domain: (,)(-\infty, \infty) or all real numbers.


Range:

The range consists of all possible values of f(x)f(x).

  • The term x2x^2 represents a squared function, which is always non-negative (x20x^2 \geq 0).
  • Adding 1 to x2x^2 shifts the graph upward by 1.
  • The minimum value occurs at x=0x = 0, where f(0)=1+02=1f(0) = 1 + 0^2 = 1.
  • As x±x \to \pm \infty, x2x^2 \to \infty, meaning f(x)f(x) \to \infty.

Thus, the function takes all values greater than or equal to 1.

Range: [1,)[1, \infty).


Would you like a graph or further explanation?

Related Questions:

  1. What is the domain and range of f(x)=x23f(x) = x^2 - 3?
  2. How does shifting affect the range of a quadratic function?
  3. What are the key characteristics of a quadratic function?
  4. How do you determine the minimum or maximum value of a quadratic function?
  5. What happens to the range if we modify f(x)=1+x2f(x) = 1 + x^2 to f(x)=x2+1f(x) = -x^2 + 1?

Tip:

For quadratic functions of the form ax2+bx+cax^2 + bx + c, the range is determined by the vertex. If a>0a > 0, the function has a minimum at the vertex. If a<0a < 0, the function has a maximum.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Domain and Range

Formulas

f(x) = 1 + x^2

Theorems

Properties of Polynomial Functions
Range of Quadratic Functions

Suitable Grade Level

Grades 7-9