Math Problem Statement

Fill out the table of values and select whether each function is odd, even, or neither. The functions provided are f(x) = 2x^2 + 3, f(x) = 4sin(x) + 1, and f(x) = 2x^5 + 3x.

Solution

The task in the image involves calculating values for different mathematical functions and determining whether each function is odd, even, or neither. Let's solve the functions for the given values of xx and then determine the nature of each function.

1. Polynomial Function: f(x)=2x2+3f(x) = 2x^2 + 3

Calculations:

  • For x=2,0,1,3,5x = -2, 0, 1, 3, 5

2. Trigonometric Function: f(x)=4sin(x)+1f(x) = 4\sin(x) + 1

Calculations:

  • For x=π,1,0,1,πx = -\pi, -1, 0, 1, \pi

3. Cubic Function: f(x)=2x5+3xf(x) = 2x^5 + 3x

Calculations:

  • For x=3,2,1,0,2x = -3, -2, -1, 0, 2

Nature Determination:

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

Let's compute the necessary values and determine the nature of these functions.Here are the calculated values for each function:

1. Polynomial Function: f(x)=2x2+3f(x) = 2x^2 + 3

  • For x=2,0,1,3,5x = -2, 0, 1, 3, 5, the values of yy are [11,3,5,21,53][11, 3, 5, 21, 53] respectively.

2. Trigonometric Function: ( f(x) = 4\sin(x)

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Math Problem Analysis

Mathematical Concepts

Functions
Odd and Even Functions
Trigonometry
Polynomials

Formulas

f(-x) = -f(x) (Odd Function Definition)
f(-x) = f(x) (Even Function Definition)
Quadratic function f(x) = 2x^2 + 3
Trigonometric function f(x) = 4sin(x) + 1
Cubic function f(x) = 2x^5 + 3x

Theorems

Odd and Even Function Theorem

Suitable Grade Level

Grades 10-12