Math Problem Statement

Fill out the table of values for three functions: f(x) = -x^6 + 3x^2, f(x) = sin(x) + 3, and f(x) = tan(x), and determine if each function is even, odd, or neither.

Solution

The problem in your image asks you to fill out the table of values for three functions and determine whether each function is even, odd, or neither. Here's a breakdown of the functions provided:

  1. f(x)=x6+3x2f(x) = -x^6 + 3x^2
  2. f(x)=sin(x)+3f(x) = \sin(x) + 3
  3. f(x)=tan(x)f(x) = \tan(x)

To determine if a function is even, odd, or neither, we need to follow these rules:

  • A function is even if f(x)=f(x)f(-x) = f(x) for all xx.
  • A function is odd if f(x)=f(x)f(-x) = -f(x) for all xx.
  • If neither condition holds, then the function is neither.

Let's start by analyzing the functions:

1. f(x)=x6+3x2f(x) = -x^6 + 3x^2

This is a polynomial function, and we can determine its symmetry by testing f(x)f(-x):

  • f(x)=(x)6+3(x)2=x6+3x2f(-x) = -(-x)^6 + 3(-x)^2 = -x^6 + 3x^2 Since f(x)=f(x)f(-x) = f(x), the function is even.

2. f(x)=sin(x)+3f(x) = \sin(x) + 3

For trigonometric functions, recall:

  • sin(x)=sin(x)\sin(-x) = -\sin(x) So,
  • f(x)=sin(x)+3=sin(x)+3f(-x) = \sin(-x) + 3 = -\sin(x) + 3 Since f(x)f(x)f(-x) \neq f(x) and f(x)f(x)f(-x) \neq -f(x), this function is neither.

3. f(x)=tan(x)f(x) = \tan(x)

For the tangent function, recall:

  • tan(x)=tan(x)\tan(-x) = -\tan(x) Thus,
  • f(x)=tan(x)=tan(x)f(-x) = \tan(-x) = -\tan(x) Since f(x)=f(x)f(-x) = -f(x), the function is odd.

Now you can complete the table

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Math Problem Analysis

Mathematical Concepts

Even and Odd Functions
Trigonometry
Polynomial Functions

Formulas

f(x) = -x^6 + 3x^2
f(x) = sin(x) + 3
f(x) = tan(x)

Theorems

A function is even if f(-x) = f(x)
A function is odd if f(-x) = -f(x)

Suitable Grade Level

Grades 10-12