In what interval are the following functions identical?
(a) `f(x)=x and phi (x)=10^(log x)`.
(b) `f(x)=sqrtx sqrt(x-1) and phi (x)=sqrt(x(x-1))`.

Answer

To determine the intervals in which the given functions are identical, we will analyze each function step by step.

### Part (a): Functions \( f(x) = x \) and \( \phi(x) = 10^{\log x} \)

1. **Identify the domain of \( f(x) \)**:
\[ f(x) = x \]
The function \( f(x) = x \) is defined for all real numbers:
\[ \text{Domain of } f(x) = (-\infty, \infty) \]

2. **Identify the domain of \( \phi(x) \)**:
\[ \phi(x) = 10^{\log x} \]
The logarithm function \( \log x \) is defined only for \( x > 0 \). Therefore, the domain of \( \phi(x) \) is:
\[ \text{Domain of } \phi(x) = (0, \infty) \]

3. **Simplify \( \phi(x) \)**:
Using properties of logarithms, we can simplify \( \phi(x) \):
\[ \phi(x) = 10^{\log x} = x \]
This holds true for \( x > 0 \).

4. **Determine the interval where both functions are identical**:
Since \( f(x) = x \) and \( \phi(x) = x \) for \( x > 0 \), the functions are identical in the interval:
\[ (0, \infty) \]

### Part (b): Functions \( f(x) = \sqrt{x} \sqrt{x-1} \) and \( \phi(x) = \sqrt{x(x-1)} \)

1. **Identify the domain of \( f(x) \)**:
\[ f(x) = \sqrt{x} \sqrt{x-1} \]
For \( f(x) \) to be defined, both \( x \geq 0 \) and \( x-1 \geq 0 \) must hold. Thus:
\[ x \geq 1 \]
Therefore, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = [1, \infty) \]

2. **Identify the domain of \( \phi(x) \)**:
\[ \phi(x) = \sqrt{x(x-1)} \]
For \( \phi(x) \) to be defined, \( x \) and \( x-1 \) must be non-negative:
\[ x \geq 1 \]
Therefore, the domain of \( \phi(x) \) is:
\[ \text{Domain of } \phi(x) = [1, \infty) \]

3. **Simplify \( f(x) \)**:
\[ f(x) = \sqrt{x} \sqrt{x-1} = \sqrt{x(x-1)} \]
Thus, we can see that:
\[ f(x) = \phi(x) \]
for all \( x \) in the domain.

4. **Determine the interval where both functions are identical**:
Since both functions are equal for \( x \geq 1 \), they are identical in the interval:
\[ [1, \infty) \]

### Summary of Results:
- For part (a), the functions are identical in the interval \( (0, \infty) \).
- For part (b), the functions are identical in the interval \( [1, \infty) \).