Determine algebraically whether the function f(y) = y^2 is even, odd, or neither. Discuss the symmetry of the function. Choose the correct answer below: A. The function is an even function. B. The function is an odd function. C. The function is neither an even function nor an odd function. Discuss the symmetry of the function. Choose the correct answer below: A. The graph of the function is symmetric about the y-axis. B. The graph of the function is symmetric about the x-axis. C. The graph of the function has no symmetry.
The Correct Answer and Explanation is:
Let’s first analyze the function f(y)=y2f(y) = y^2f(y)=y2 to determine whether it is even, odd, or neither.
Step 1: Check if the function is even, odd, or neither
To determine the parity of the function, we need to check the following properties:
- Even function: A function is even if f(−y)=f(y)f(-y) = f(y)f(−y)=f(y) for all values of yyy.
- Odd function: A function is odd if f(−y)=−f(y)f(-y) = -f(y)f(−y)=−f(y) for all values of yyy.
Now, for the given function f(y)=y2f(y) = y^2f(y)=y2:
- f(−y)=(−y)2=y2f(-y) = (-y)^2 = y^2f(−y)=(−y)2=y2.
Since f(−y)=f(y)f(-y) = f(y)f(−y)=f(y) for all values of yyy, the function satisfies the condition for being even.
Step 2: Analyze the symmetry of the graph
For even functions, the graph is symmetric about the y-axis. This is because f(y)=f(−y)f(y) = f(-y)f(y)=f(−y), meaning the values of the function at yyy and −y-y−y are identical. The function f(y)=y2f(y) = y^2f(y)=y2 will look the same on both sides of the y-axis.
Thus, the symmetry of the graph is about the y-axis.
Conclusion:
- The function f(y)=y2f(y) = y^2f(y)=y2 is even.
- The graph of the function is symmetric about the y-axis.
Therefore, the correct answers are:
- A. The function is an even function.
- A. The graph of the function is symmetric about the y-axis.
