A function whose graph is symmetrical about y-axis is

Options

A. `f(x)=x((3^(x)-1)/(3^(x)-1))`
B. `f(x)=log_(2)(x+sqrt(x^(2)+1))`
C. `f(x+y)=f(x)+f(y)`
D. `f(x)=sinx+cosx`

Explanation

To determine which function is symmetrical about the y-axis, we need to identify the properties of even functions. An even function satisfies the condition:

\[ f(-x) = f(x) \]

This means that the function's output for \( -x \) is the same as its output for \( x \).

### Step-by-Step Solution:

1. **Understanding the Condition for Even Functions**:
- A function is even if its graph is symmetrical about the y-axis. This can be mathematically expressed as:
\[ f(-x) = f(x) \]

2. **Analyzing the Given Function**:
- We need to analyze the first option given in the question. Let's denote the function as:
\[ f(x) = \frac{3^{-x} - 1}{3^x + 1} \]

3. **Finding \( f(-x) \)**:
- We will compute \( f(-x) \):
\[ f(-x) = \frac{3^{-(-x)} - 1}{3^{-x} + 1} = \frac{3^{x} - 1}{3^{-x} + 1} \]

4. **Simplifying \( f(-x) \)**:
- The denominator can be rewritten using the property of exponents:
\[ 3^{-x} = \frac{1}{3^x} \]
- Thus, we have:
\[ f(-x) = \frac{3^{x} - 1}{\frac{1}{3^x} + 1} = \frac{3^{x} - 1}{\frac{1 + 3^x}{3^x}} = \frac{(3^{x} - 1) \cdot 3^x}{1 + 3^x} \]

5. **Finding \( f(x) \)**:
- Now, let's write down \( f(x) \) again for comparison:
\[ f(x) = \frac{3^{-x} - 1}{3^x + 1} \]

6. **Comparing \( f(-x) \) and \( f(x) \)**:
- We need to check if \( f(-x) = f(x) \). After simplification, we can see if both expressions are identical.

7. **Conclusion**:
- If \( f(-x) \) simplifies to \( f(x) \), then the function is even and thus symmetrical about the y-axis.

### Final Answer:
The function whose graph is symmetrical about the y-axis is an even function.