Which of the following functions is an odd functions ?

Options

A. `f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2))`
B. `f(x)=x((a^(x)+1)/(a^(x)-1))`
C. `f(x)=log_(10)((1-x^(2))/(1+x^(2)))`
D. f(x)=k (constant )

Explanation

To determine which of the given functions is an odd function, we need to recall the definition of an odd function. A function \( f(x) \) is considered an odd function if it satisfies the condition:

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

We will analyze each option one by one to see if they meet this criterion.

### Step 1: Analyze Option 1
Let’s denote the first function as \( f(x) = x^3 + x \).

Now, we will calculate \( f(-x) \):

\[ f(-x) = (-x)^3 + (-x) = -x^3 - x \]

Now, we check if \( f(-x) = -f(x) \):

\[ -f(x) = -(x^3 + x) = -x^3 - x \]

Since \( f(-x) = -f(x) \), Option 1 is an odd function.

### Step 2: Analyze Option 2
Let’s denote the second function as \( f(x) = x^2 + 1 \).

Now, we will calculate \( f(-x) \):

\[ f(-x) = (-x)^2 + 1 = x^2 + 1 \]

Now, we check if \( f(-x) = -f(x) \):

\[ -f(x) = -(x^2 + 1) = -x^2 - 1 \]

Since \( f(-x) \neq -f(x) \), Option 2 is not an odd function.

### Step 3: Analyze Option 3
Let’s denote the third function as \( f(x) = \log(1 - x^2) \).

Now, we will calculate \( f(-x) \):

\[ f(-x) = \log(1 - (-x)^2) = \log(1 - x^2) \]

Now, we check if \( f(-x) = -f(x) \):

\[ -f(x) = -\log(1 - x^2) \]

Since \( f(-x) \neq -f(x) \), Option 3 is not an odd function.

### Step 4: Analyze Option 4
Let’s denote the fourth function as \( f(x) = k \) (a constant function).

Now, we will calculate \( f(-x) \):

\[ f(-x) = k \]

Now, we check if \( f(-x) = -f(x) \):

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

Since \( f(-x) \neq -f(x) \), Option 4 is not an odd function.

### Conclusion
The only function that satisfies the condition of being an odd function is **Option 1**.