A function f is said to be even, if

Options

A. `f (x) =-f (x)`
B. `f(-x) =f(x)`
C. `f (-x) =-f (x)`
D. none of these

Answer

To determine when a function \( f \) is considered even, we follow these steps:

### Step-by-Step Solution:

1. **Definition of an Even Function**: A function \( f \) is defined as even if it satisfies the condition:
\[ f(-x) = f(x) \quad \text{for all } x \text{ in the domain of } f. \]

2. **Symmetry with Respect to the Y-Axis**: The graph of an even function is symmetric with respect to the y-axis. This means that for every point \( (x, f(x)) \) on the graph, there is a corresponding point \( (-x, f(-x)) \) that is also on the graph, and these points have the same y-coordinate.

3. **Examples**: Common examples of even functions include:
- \( f(x) = x^2 \)
- \( f(x) = \cos(x) \)
- \( f(x) = x^4 \)

For these functions, if you substitute \( -x \) into the function, you will get the same output as when you substitute \( x \).

4. **Verification**: To verify if a function is even, you can:
- Take any value of \( x \).
- Calculate \( f(x) \) and \( f(-x) \).
- If both results are equal, then the function is even.

5. **Conclusion**: Thus, a function \( f \) is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain.