Problem 22
Question
State whether each of the following is an odd function, an even function, or neither. Prove your statements. (a) The sum of two even functions (b) The sum of two odd functions (c) The product of two even functions (d) The product of two odd functions (e) The product of an even function and an odd function
Step-by-Step Solution
Verified Answer
(a) Even, (b) Odd, (c) Even, (d) Even, (e) Odd.
1Step 1: Recap Definitions
An \textbf{odd function} satisfies the condition \( f(-x) = -f(x) \) for all \( x \). An \textbf{even function} satisfies \( f(-x) = f(x) \). If neither condition holds, the function is neither odd nor even.
2Step 2: Sum of Two Even Functions
Let \( f(x) \) and \( g(x) \) be even functions. Then \( f(-x) = f(x) \) and \( g(-x) = g(x) \). Consider their sum \( h(x) = f(x) + g(x) \). Then \( h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x) \). Thus, the sum of two even functions is an even function.
3Step 3: Sum of Two Odd Functions
Let \( f(x) \) and \( g(x) \) be odd functions. Then \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). Consider their sum \( h(x) = f(x) + g(x) \). Then \( h(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \). Thus, the sum of two odd functions is an odd function.
4Step 4: Product of Two Even Functions
Let \( f(x) \) and \( g(x) \) be even functions. Then \( f(-x) = f(x) \) and \( g(-x) = g(x) \). Consider their product \( h(x) = f(x)g(x) \). Then \( h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) \). Therefore, the product of two even functions is an even function.
5Step 5: Product of Two Odd Functions
Let \( f(x) \) and \( g(x) \) be odd functions. Then \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). Consider their product \( h(x) = f(x)g(x) \). Then \( h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x) \). Hence, the product of two odd functions is an even function.
6Step 6: Product of an Even and an Odd Function
Let \( f(x) \) be an even function and \( g(x) \) be an odd function. Then \( f(-x) = f(x) \) and \( g(-x) = -g(x) \). Consider their product \( h(x) = f(x)g(x) \). Then \( h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x) \). Therefore, the product of an even and an odd function is an odd function.
Key Concepts
Function PropertiesSum of FunctionsProduct of FunctionsMathematical Proofs
Function Properties
Understanding the properties of functions is crucial in mathematics. Two important types of functions are **even** and **odd** functions.
An **even function** has the property that it is symmetric about the y-axis. This means for every x, the function satisfies the condition: \( f(-x) = f(x) \). In other words, if you flip the graph of the function around the y-axis, it remains unchanged. Common examples of even functions include quadratic functions like \( f(x) = x^2 \).
Odd functions, on the other hand, have rotational symmetry about the origin. This means if you rotate the graph of an odd function 180 degrees around the origin, it looks the same. Mathematically, this is expressed as: \( f(-x) = -f(x) \). A classic example is the cubic function \( f(x) = x^3 \).
When a function doesn't satisfy either condition, it is neither even nor odd. Recognizing these properties helps us understand more about the behavior and symmetries in functions.
An **even function** has the property that it is symmetric about the y-axis. This means for every x, the function satisfies the condition: \( f(-x) = f(x) \). In other words, if you flip the graph of the function around the y-axis, it remains unchanged. Common examples of even functions include quadratic functions like \( f(x) = x^2 \).
Odd functions, on the other hand, have rotational symmetry about the origin. This means if you rotate the graph of an odd function 180 degrees around the origin, it looks the same. Mathematically, this is expressed as: \( f(-x) = -f(x) \). A classic example is the cubic function \( f(x) = x^3 \).
When a function doesn't satisfy either condition, it is neither even nor odd. Recognizing these properties helps us understand more about the behavior and symmetries in functions.
Sum of Functions
When adding functions together, we must consider their properties to understand the type of function the sum will produce.
- **Sum of Two Even Functions**: Let's consider two even functions \( f(x) \) and \( g(x) \). Since they both satisfy \( f(-x) = f(x) \) and \( g(-x) = g(x) \), their sum \( h(x) = f(x) + g(x) \) will also be even because: \[ h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x) \] This means the sum inherits the symmetry about the y-axis.
- **Sum of Two Odd Functions**: For two odd functions \( f(x) \) and \( g(x) \), each satisfies \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). Their sum \( h(x) = f(x) + g(x) \) turns out to be odd as well because: \[ h(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \] Thus the resulting function maintains rotational symmetry about the origin.
Understanding these operations helps predict the nature of new functions derived from basic building blocks.
- **Sum of Two Even Functions**: Let's consider two even functions \( f(x) \) and \( g(x) \). Since they both satisfy \( f(-x) = f(x) \) and \( g(-x) = g(x) \), their sum \( h(x) = f(x) + g(x) \) will also be even because: \[ h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x) \] This means the sum inherits the symmetry about the y-axis.
- **Sum of Two Odd Functions**: For two odd functions \( f(x) \) and \( g(x) \), each satisfies \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). Their sum \( h(x) = f(x) + g(x) \) turns out to be odd as well because: \[ h(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \] Thus the resulting function maintains rotational symmetry about the origin.
Understanding these operations helps predict the nature of new functions derived from basic building blocks.
Product of Functions
The product of functions interacts differently compared to the sum, due to the multiplicative properties of even and odd functions.
- **Product of Two Even Functions**: When multiplying two even functions \( f(x) \) and \( g(x) \), each retaining \( f(-x) = f(x) \) and \( g(-x) = g(x) \), their product \( h(x) = f(x)g(x) \) remains even: \[ h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) \] This simply means the resultant function continues to reflect y-axis symmetry.
- **Product of Two Odd Functions**: If both functions are odd, \( f(x) \) and \( g(x) \), then their product \( h(x) = f(x)g(x) \) becomes even:\[ h(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x) \] This occurs because the negatives effectively cancel out, resulting in a function symmetrical about the y-axis.
- **Product of an Even and an Odd Function**: Combining an even function \( f(x) \) and an odd function \( g(x) \), their product \( h(x) = f(x)g(x) \) results in an odd function:\[ h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x) \] These manipulations illustrate how mixing the characteristics of functions affects the overall properties.
- **Product of Two Even Functions**: When multiplying two even functions \( f(x) \) and \( g(x) \), each retaining \( f(-x) = f(x) \) and \( g(-x) = g(x) \), their product \( h(x) = f(x)g(x) \) remains even: \[ h(-x) = f(-x)g(-x) = f(x)g(x) = h(x) \] This simply means the resultant function continues to reflect y-axis symmetry.
- **Product of Two Odd Functions**: If both functions are odd, \( f(x) \) and \( g(x) \), then their product \( h(x) = f(x)g(x) \) becomes even:\[ h(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x) \] This occurs because the negatives effectively cancel out, resulting in a function symmetrical about the y-axis.
- **Product of an Even and an Odd Function**: Combining an even function \( f(x) \) and an odd function \( g(x) \), their product \( h(x) = f(x)g(x) \) results in an odd function:\[ h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x) \] These manipulations illustrate how mixing the characteristics of functions affects the overall properties.
Mathematical Proofs
Mathematical proofs form the foundation for establishing truths within mathematics. When proving properties about functions, it’s essential to rely on definitions and analytical reasoning.
To **prove** a function is even, demonstrate explicitly that \( f(-x) = f(x) \) holds for all x. Confirmation requires showing the entire function graph mirrors across the y-axis.
To prove a function is odd, you need to verify that \( f(-x) = -f(x) \) universally holds. This means showing that for every point on the function graph, rotating it 180 degrees about the origin results in the same graph.
Proofs rely heavily on logical steps with each statement built upon previously verified truths. When working with function sums or products, leverage fundamental definitions to conclude properties such as symmetry and rotation.
Always approach proofs by simplifying complex terms through substitution and ensure each step logically follows the last before concluding.
To **prove** a function is even, demonstrate explicitly that \( f(-x) = f(x) \) holds for all x. Confirmation requires showing the entire function graph mirrors across the y-axis.
To prove a function is odd, you need to verify that \( f(-x) = -f(x) \) universally holds. This means showing that for every point on the function graph, rotating it 180 degrees about the origin results in the same graph.
Proofs rely heavily on logical steps with each statement built upon previously verified truths. When working with function sums or products, leverage fundamental definitions to conclude properties such as symmetry and rotation.
Always approach proofs by simplifying complex terms through substitution and ensure each step logically follows the last before concluding.
Other exercises in this chapter
Problem 22
in Problems 17-22, find the center and radius of the circle with the given equation. \(x^{2}+16 x+\frac{105}{16}+4 y^{2}+3 y=0\)
View solution Problem 22
$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{4}\left(\frac{1}{2 x}\right)=3 $$
View solution Problem 22
In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=\frac{x}{x^{2}+1}
View solution Problem 22
In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(\phi(z)=\frac{2 z+1}{z-1}\)
View solution