`f(x)=sqrt(1-x^(2)), g(x)=sqrt(1-x)*sqrt(1+x)` . Identical functions or not?

Answer

To determine if the functions \( f(x) = \sqrt{1 - x^2} \) and \( g(x) = \sqrt{1 - x} \cdot \sqrt{1 + x} \) are identical, we will follow the steps outlined in the video transcript, which involves checking the domain, range, and equality of the functions over their common domain.

### Step 1: Determine the Domain of \( f(x) \)

1. The function \( f(x) = \sqrt{1 - x^2} \) is defined when the expression inside the square root is non-negative:
\[ 1 - x^2 \geq 0 \]
This can be rearranged to:
\[ x^2 \leq 1 \]
Taking the square root gives:
\[ -1 \leq x \leq 1 \]
Thus, the domain of \( f(x) \) is:
\[ D_f = [-1, 1] \]

### Step 2: Determine the Domain of \( g(x) \)

2. The function \( g(x) = \sqrt{1 - x} \cdot \sqrt{1 + x} \) is defined when both square roots are non-negative:
- For \( \sqrt{1 - x} \):
\[ 1 - x \geq 0 \implies x \leq 1 \]
- For \( \sqrt{1 + x} \):
\[ 1 + x \geq 0 \implies x \geq -1 \]
Combining these inequalities gives:
\[ -1 \leq x \leq 1 \]
Thus, the domain of \( g(x) \) is:
\[ D_g = [-1, 1] \]

### Step 3: Check if the Domains are Equal

3. Since both functions have the same domain:
\[ D_f = D_g = [-1, 1] \]

### Step 4: Check the Range of \( f(x) \)

4. The range of \( f(x) = \sqrt{1 - x^2} \) can be found by observing that as \( x \) varies from -1 to 1, \( f(x) \) takes values from 0 to 1:
- At \( x = -1 \) and \( x = 1 \), \( f(x) = 0 \).
- At \( x = 0 \), \( f(x) = 1 \).
Thus, the range of \( f(x) \) is:
\[ R_f = [0, 1] \]

### Step 5: Check the Range of \( g(x) \)

5. The function \( g(x) = \sqrt{1 - x} \cdot \sqrt{1 + x} \) can be simplified:
\[ g(x) = \sqrt{(1 - x)(1 + x)} = \sqrt{1 - x^2} \]
Therefore, the range of \( g(x) \) is the same as \( f(x) \):
\[ R_g = [0, 1] \]

### Step 6: Check if the Ranges are Equal

6. Since both functions have the same range:
\[ R_f = R_g = [0, 1] \]

### Step 7: Check if \( f(x) = g(x) \) for all \( x \) in the Domain

7. We already simplified \( g(x) \) to:
\[ g(x) = \sqrt{1 - x^2} \]
Therefore, for all \( x \) in the domain:
\[ f(x) = g(x) \]

### Conclusion

Since the domain, range, and values of the functions are equal for all \( x \) in the common domain, we conclude that:
\[ f(x) \text{ and } g(x) \text{ are identical functions.} \]