Determine algebraically whether or not the function is even or odd, and justify your answer. (1 point) Responses The function is odd because . The function is odd because Image with alt text: f left parenthesis negative x right parenthesis equals negative f left parenthesis x right parenthesis . The function is odd because . The function is odd because Image with alt text: f left parenthesis negative x right parenthesis equals f left parenthesis x right parenthesis . The function is even because . The function is even because Image with alt text: f left parenthesis negative x right parenthesis equals negative f left parenthesis x right parenthesis . The function is even because . The function is even because Image with alt text: f left parenthesis negative x right parenthesis equals f left parenthesis x right parenthesis . The function is neither even nor odd because and .
The function is odd because f(-x) = -f(x).
To determine whether a function is even or odd, we need to examine the algebraic expression of the function.
1. An even function has the property that f(-x) = f(x) for all x in the domain of the function.
2. An odd function has the property that f(-x) = -f(x) for all x in the domain of the function.
In the given responses, there are four possibilities: odd, odd, even, and even. To determine the correct answer, we need to evaluate the given function symbolically.
As the given responses are not complete and only provide incomplete statements like "The function is odd because..." or "The function is even because...", it is not possible to determine the correct answer algebraically without the actual function expression.
To determine if the function is even or odd, we need the actual algebraic expression of the function.
To determine whether a function is even or odd algebraically, we need to examine its equation.
For a function to be even, it must satisfy the property: f(-x) = f(x) for all x in the domain.
For a function to be odd, it must satisfy the property: f(-x) = -f(x) for all x in the domain.
To determine if the function is even or odd, we can substitute -x into the function and check if the resulting expression is equal to the original function or its negative.
Let's examine the choices:
- If the function satisfies f(-x) = -f(x), it is odd.
- If the function satisfies f(-x) = f(x), it is even.
- If the function does not satisfy either property, it is neither even nor odd.
You are given several response options, each providing different explanations and/or images. Since I cannot view or interpret any images, I will focus on the statements alone:
1. "The function is odd because ."
Since the statement is incomplete, it does not provide sufficient information to determine whether the function is odd.
2. "The function is odd because f(-x) = -f(x)."
This statement correctly identifies the function as odd. When substituting -x into the function, we get the negative of the original function.
3. "The function is odd because ."
Similar to statement 1, this incomplete statement does not provide sufficient information to determine the nature of the function.
4. "The function is odd because f(-x) = f(x)."
This statement is incorrect. If the function satisfies f(-x) = f(x), it would be even, not odd.
5. "The function is even because ."
Similar to statement 1 and 3, this incomplete statement does not provide enough information.
6. "The function is even because f(-x) = -f(x)."
This statement is incorrect. If the function satisfies f(-x) = -f(x), it would be odd, not even.
7. "The function is even because f(-x) = f(x)."
This statement correctly identifies the function as even. When substituting -x into the function, we obtain the same expression as the original function.
8. "The function is even because ."
Similar to statement 1, 3, and 5, this statement is incomplete and does not provide sufficient information.
Based on the explanations provided, we can conclude that the function is odd if f(-x) = -f(x) and even if f(-x) = f(x). If neither property is satisfied, the function is neither even nor odd.