Math Problem Statement

Determine whether the following functions are even or odd:

Solution

To determine whether the function f(x)=xx2+5f(x) = x \sqrt{x^2 + 5} is even, odd, or neither, we will use the following definitions:

  1. Even Function: A function f(x)f(x) is even if f(x)=f(x)f(-x) = f(x) for all xx.
  2. Odd Function: A function f(x)f(x) is odd if f(x)=f(x)f(-x) = -f(x) for all xx.

Let's test these properties.

Step 1: Compute f(x)f(-x)

The given function is: f(x)=xx2+5f(x) = x \sqrt{x^2 + 5} Now, substitute x-x into the function: f(x)=(x)(x)2+5=(x)x2+5f(-x) = (-x) \sqrt{(-x)^2 + 5} = (-x) \sqrt{x^2 + 5}

Step 2: Compare f(x)f(-x) with f(x)f(x)

We have: f(x)=xx2+5f(-x) = -x \sqrt{x^2 + 5} And: f(x)=xx2+5f(x) = x \sqrt{x^2 + 5}

  • f(x)f(x)f(-x) \neq f(x), so the function is not even.
  • f(x)=f(x)f(-x) = -f(x), which shows the function satisfies the condition for being odd.

Conclusion:

The function f(x)=xx2+5f(x) = x \sqrt{x^2 + 5} is odd.

Would you like further clarification or details on this? Here are some related questions to explore this concept further:

  1. What are some examples of even functions?
  2. Can a function be neither even nor odd? If so, how can we identify it?
  3. How would you check if a polynomial function is even or odd?
  4. Can the square root function x\sqrt{x} be even or odd?
  5. What are the applications of even and odd functions in physics or engineering?

Tip: When determining whether a function is even or odd, always check both conditions f(x)=f(x)f(-x) = f(x) and f(x)=f(x)f(-x) = -f(x) systematically to avoid mistakes!

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Math Problem Analysis

Mathematical Concepts

Functions
Even Functions
Odd Functions

Formulas

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Theorems

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Suitable Grade Level

Grades 11-12