Math Problem Statement

Based on the ordered pairs seen in the table, make a conjecture about whether the function is even, odd, or neither. Provide clear reasoning.

Solution

The table provided shows several ordered pairs (x,f(x))(x, f(x)) for a function f(x)f(x):

xxf(x)f(x)
3-39
2-25
1-1-1
00
1-1
25
39

To determine whether the function is even, odd, or neither, we need to check the symmetry properties:

  1. Even Function: A function is even if f(x)=f(x)f(x) = f(-x) for all xx. This means that the function is symmetric with respect to the y-axis.
  2. Odd Function: A function is odd if f(x)=f(x)f(x) = -f(-x) for all xx. This means that the function is symmetric with respect to the origin.

Analysis of the Given Values:

  • For x=3x = -3, f(3)=9f(-3) = 9 and for x=3x = 3, f(3)=9f(3) = 9. This satisfies f(x)=f(x)f(x) = f(-x).
  • For x=2x = -2, f(2)=5f(-2) = 5 and for x=2x = 2, f(2)=5f(2) = 5. This also satisfies f(x)=f(x)f(x) = f(-x).
  • For x=1x = -1, f(1)=1f(-1) = -1 and for x=1x = 1, f(1)=1f(1) = -1. Again, f(x)=f(x)f(x) = f(-x) is satisfied.
  • For x=0x = 0, f(0)=0f(0) = 0.

Since for every xx, f(x)=f(x)f(x) = f(-x), the function is even.

Conclusion:

The function is even because for every value of xx given, the function satisfies f(x)=f(x)f(x) = f(-x).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

  1. How do you determine if a function is odd using its graph?
  2. What are the characteristics of a function that is neither even nor odd?
  3. Can a function be both even and odd? If so, under what conditions?
  4. How do transformations like translations affect the evenness or oddness of a function?
  5. How can you use calculus to verify if a function is even or odd?

Tip: To check if a function is even or odd, plug in x-x and compare it to f(x)f(x). If f(x)=f(x)f(-x) = f(x), the function is even; if f(x)=f(x)f(-x) = -f(x), the function is odd.

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Math Problem Analysis

Mathematical Concepts

Function Analysis
Symmetry in Functions

Formulas

Even Function: f(x) = f(-x)
Odd Function: f(x) = -f(-x)

Theorems

Definition of Even and Odd Functions

Suitable Grade Level

Grades 8-10