Problem 20 An even function satisfies the p... [FREE SOLUTION]

Chapter 10: Problem 20

An even function satisfies the property \(f(-x)=f(x)\) for all \(x\) in the domain of \(f(x)\) An odd function satisfies the property \(f(-x)=-f(x)\) for all \(x\) in the domain of \(f(x)\). Devise and test an algebraic method to determine if the sum of two functions is even, odd, or neither. Show by example how your method works. Use at least three of the functions you have studied: absolute value, radical, polynomial, trigonometric, exponential, logarithmic, and rational.

Short Answer

Expert verified

The sum of an even and an odd function is neither even nor odd. The sum of two odd functions is odd. The sum of an absolute value and a sine function is neither even nor odd.

Step by step solution

Understand the Definitions

Review the definitions of even and odd functions: Even function: \(f(-x) = f(x)\) Odd function: \(f(-x) = -f(x)\).

Define Functions for Testing

Choose three types of functions to test. For this example, choose the absolute value function \(g(x) = |x|\), the polynomial function \(h(x) = x^3\), and the trigonometric function \(k(x) = \sin(x)\).

Sum Two Functions

First, test the sum of \(g(x)\) and \(h(x)\): \ Define \(m(x) = g(x) + h(x) = |x| + x^3\).

Test Even or Odd for Sum of \(g(x)\) and \(h(x)\)

Compute \(m(-x)\) and compare it to \(m(x)\): \ \(m(-x) = | -x | + (-x)^3 = |x| - x^3\). \ \(m(-x) eq m(x)\) and \(m(-x) eq -m(x)\). Therefore, \(m(x) = |x| + x^3\) is neither even nor odd.

Sum Another Two Functions

Next, test the sum of \(h(x)\) and \(k(x)\): \ Define \(n(x) = h(x) + k(x) = x^3 + \sin(x)\).

Test Even or Odd for Sum of \(h(x)\) and \(k(x)\)

Compute \(n(-x)\) and compare it to \(n(x)\): \ \(n(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin(x)\). \ \(n(-x) = -n(x)\). Therefore, \(n(x) = x^3 + \sin(x)\) is an odd function.

Sum Final Two Functions

Lastly, test the sum of \(g(x)\) and \(k(x)\): \ Define \(p(x) = g(x) + k(x) = |x| + \sin(x)\).

Test Even or Odd for Sum of \(g(x)\) and \(k(x)\)

Compute \(p(-x)\) and compare it to \(p(x)\): \ \(p(-x) = | -x | + \sin(-x) = |x| - \sin(x)\). \ \(p(-x) eq p(x)\) and \(p(-x) eq -p(x)\). Therefore, \(p(x) = |x| + \sin(x)\) is neither even nor odd.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

even function

An even function is a function that satisfies the property: ```math f(-x) = f(x) ``` for all x in its domain. This means that the function's graph is symmetrical with respect to the y-axis. A classic example of an even function is the absolute value function ```math abs(x)``` or a polynomial function like ```math x^2``` . If you were to fold the graph along the y-axis, both halves would match perfectly.
To identify an even function, simply apply the negative input and see if the function's value remains unchanged.

odd function

An odd function satisfies the property: ```math f(-x) = -f(x) ``` for all x in its domain. This means that the function's graph is symmetrical with respect to the origin. To visualize this, you can rotate the graph 180 degrees around the origin and it will look the same. A typical example of an odd function is ```math x^3 ``` or the sine function ```math sin(x) ``` . To test if a function is odd, substitute a negative input and check if the result is the negative of the original function's value.

function properties

Understanding the properties of functions is crucial in determining whether they are even, odd, or neither. These properties include:
- **Symmetry**: Even functions exhibit y-axis symmetry, while odd functions exhibit origin symmetry.
- **Algebraic Tests**: For even functions, ```math f(-x) = f(x) ``` . For odd functions, ```math f(-x) = -f(x) ``` . If neither condition holds, the function is neither even nor odd.
- **Graphical Analysis**: Plotting the function can also help to visually identify symmetry properties.
Exploratory methods like summing different functions, as discussed in the exercise above, can also expose the inherent symmetry properties.

polynomial function

Polynomial functions are algebraic expressions composed of variables and coefficients, involving only non-negative integer exponents of variables. General polynomials take the form: ```math a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 ```.
Examples include ```math x^2 and ```math x^3 . Whether a polynomial function is even or odd depends on the exponents of its terms.
- **Even Polynomial**: All exponents are even. Example: ```math x^2 .
- **Odd Polynomial**: All exponents are odd. Example: ```math x^3```.
In mixed polynomials with both even and odd terms, the function will be neither even nor odd.

trigonometric function

Trigonometric functions like sine and cosine also exhibit even and odd properties. - **Sine Function**: The function ```math sin(x)``` is odd because ```math sin(-x) = -sin(x) ```. This implies origin symmetry.
- **Cosine Function**: The function ```math cos(x)``` is even because ```math cos(-x)= cos(x) ```. This implies y-axis symmetry.
Understanding these properties helps in determining the overall symmetry of composed equations involving trigonometric functions.

absolute value function

The absolute value function, denoted by ```math |x|``` is a classic example of an even function. This is because: ```math |-x|=|x|.``` Regardless of whether the input is positive or negative, the absolute value function always outputs a non-negative value, creating y-axis symmetry. In summary:
- **Definition**: ```math |x|``` returns x if x is positive, and -x if x is negative.
- **Properties**: It is always even and produces non-negative outputs.
This makes it useful in various mathematical scenarios requiring non-negative results.

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