Math Problem Statement

What is the range of f(x)=x 2 for all real x ?

Question 2Answer A. (−∞,0]

B. [0,1]

C. (−∞,∞)

D. [0,∞)

Solution

To determine the range of the function f(x)=x2f(x) = x^2 for all real xx:

Step-by-step Solution:

  1. Definition of f(x)=x2f(x) = x^2:

    • This is a quadratic function that outputs the square of any real number xx.
  2. Behavior of x2x^2:

    • For any real xx, x20x^2 \geq 0.
    • Negative values for f(x)f(x) are not possible because squaring a number always results in a non-negative value.
  3. Minimum value:

    • The minimum value of f(x)f(x) is 00, which occurs when x=0x = 0.
  4. Maximum value:

    • There is no maximum value; as xx \to \infty or xx \to -\infty, f(x)f(x) \to \infty.
  5. Range:

    • The set of all possible values of f(x)f(x) is [0,)[0, \infty), which includes 00 and all positive numbers.

Final Answer:

D. [0, ∞)


Would you like further explanation or visualizations?
Here are some related questions for practice:

  1. What is the domain of f(x)=x2f(x) = x^2?
  2. Find the range of f(x)=(x3)2+1f(x) = (x - 3)^2 + 1.
  3. How does the range of x2x^2 change if the domain is restricted to x0x \geq 0?
  4. What is the range of f(x)=x2f(x) = -x^2?
  5. Determine the range of f(x)=x2+2x+1f(x) = x^2 + 2x + 1.

Tip: To find the range of a function, analyze its behavior as x,xx \to \infty, x \to -\infty, and its critical points.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Quadratic Functions
Range of a Function

Formulas

f(x) = x^2

Theorems

Properties of Quadratic Functions
Range Analysis for Functions

Suitable Grade Level

Grades 9-10