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Let's break down the problem in the image.

You are given that the function \( f(x) \) is defined over the domain \( [0,1] \), and you are asked to find the domain for the following two cases:

1. \( f(\sin x) \)
2. \( g(x) = f(x + a) + f(x - a) \)

### 1. Domain of \( f(\sin x) \):
Since the domain of \( f(x) \) is given as \( [0, 1] \), we need to ensure that the input of \( f \), which is \( \sin x \), also lies within the interval \( [0, 1] \).

We know that the sine function \( \sin x \) takes values in the range \( [-1, 1] \) for all real numbers \( x \). Therefore, for \( f(\sin x) \) to be well-defined, \( \sin x \) must lie within the domain \( [0, 1] \) (the domain of \( f(x) \)).

Thus, we require:
\[
0 \leq \sin x \leq 1
\]
This inequality is satisfied when \( x \in [0, \pi] \). Therefore, the domain of \( f(\sin x) \) is \( [0, \pi] \).

### 2. Domain of \( g(x) = f(x + a) + f(x - a) \):
For \( g(x) \), we need to ensure that both \( f(x + a) \) and \( f(x - a) \) are defined. Since \( f(x) \) is defined on \( [0, 1] \), the arguments \( x + a \) and \( x - a \) must both lie within this interval.

Thus, we require:
\[
0 \leq x + a \leq 1 \quad \text{and} \quad 0 \leq x - a \leq 1
\]

From the first inequality \( 0 \leq x + a \leq 1 \), we get:
\[
-x \leq a \leq 1 - x
\]

From the second inequality \( 0 \leq x - a \leq 1 \), we get:
\[
a \leq x \leq a + 1
\]

The domain of \( g(x) \) depends on the value of \( a \), but generally, we must find the overlap of the two intervals that satisfy both conditions.

---

Would you like more details on solving part 2 or explanations on handling the inequalities? Here are some related questions for deeper understanding:

1. How do we solve trigonometric inequalities involving functions like sine or cosine?
2. What are common techniques to find the domain of composite functions?
3. How does the domain of a function restrict the domain of compositions?
4. How would changing the domain of \( f(x) \) from \( [0, 1] \) to another interval affect the result?
5. Can the value of \( a \) in part 2 make the function \( g(x) \) undefined for certain \( x \)?

**Tip:** Always check the range of the inner function when finding the domain of composite functions like \( f(\sin x) \).

设函数 y=f(x) 的定义域为 [0, 1], 求下列函数的定义域: 1) f(sin x); 2) g(x) = f(x + a) + f(x - a)

Explore how to find the domain of the functions f(sin x) and g(x) = f(x + a) + f(x - a) when f(x) is defined over [0, 1]. Learn the steps involving trigonometric functions and composite functions, and understand how inequalities help define the solution. Suitable for advanced high school students.

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