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The given problem asks to find the first-degree function \( f(x) \) that satisfies the following conditions:

\[
\int_{-1}^1 f(x) \, dx = -1,
\]
\[
\int_0^1 x f(x) \, dx = 0.
\]

### Step 1: Assume the form of \( f(x) \)
Since \( f(x) \) is a first-degree (linear) function, we assume:
\[
f(x) = ax + b
\]
where \( a \) and \( b \) are constants to be determined.

### Step 2: Apply the first condition
We know:
\[
\int_{-1}^1 f(x) \, dx = -1.
\]
Substitute \( f(x) = ax + b \):
\[
\int_{-1}^1 (ax + b) \, dx = -1.
\]
This can be split into two integrals:
\[
\int_{-1}^1 ax \, dx + \int_{-1}^1 b \, dx = -1.
\]
First, evaluate the integrals:
\[
\int_{-1}^1 ax \, dx = a \int_{-1}^1 x \, dx = a \left[\frac{x^2}{2}\right]_{-1}^1 = a \left( \frac{1}{2} - \frac{1}{2} \right) = 0,
\]
\[
\int_{-1}^1 b \, dx = b \int_{-1}^1 dx = b [x]_{-1}^1 = b(1 - (-1)) = 2b.
\]
So the first condition simplifies to:
\[
2b = -1 \quad \Rightarrow \quad b = -\frac{1}{2}.
\]

### Step 3: Apply the second condition
Now use the second condition:
\[
\int_0^1 x f(x) \, dx = 0.
\]
Substitute \( f(x) = ax - \frac{1}{2} \):
\[
\int_0^1 x(ax - \frac{1}{2}) \, dx = 0.
\]
This can be expanded to:
\[
\int_0^1 (ax^2 - \frac{1}{2}x) \, dx = 0.
\]
Now, evaluate the integrals:
\[
\int_0^1 ax^2 \, dx = a \int_0^1 x^2 \, dx = a \left[ \frac{x^3}{3} \right]_0^1 = a \cdot \frac{1}{3},
\]
\[
\int_0^1 \frac{1}{2}x \, dx = \frac{1}{2} \int_0^1 x \, dx = \frac{1}{2} \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}.
\]
So the second condition simplifies to:
\[
a \cdot \frac{1}{3} - \frac{1}{4} = 0.
\]
Solve for \( a \):
\[
a = \frac{1}{4} \cdot \frac{3}{1} = \frac{3}{4}.
\]

### Step 4: Final answer
Thus, the function \( f(x) \) that satisfies both conditions is:
\[
f(x) = \frac{3}{4}x - \frac{1}{2}.
\]

Would you like further clarification on any part of this solution? Here are some related questions to deepen your understanding:

1. What is the geometric interpretation of the integral conditions on \( f(x) \)?
2. How would the solution change if the conditions were modified?
3. What is the significance of the function being linear in this context?
4. Can we solve this problem using a different method, such as a system of equations?
5. What happens if we integrate the function over a different range?

**Tip**: Always double-check your integrals, especially when combining linear terms, as errors can propagate.

Find the first-degree function f(x) that satisfies the following conditions:
\int_{-1}^1 f(x) dx = -1, \int_0^1 x f(x) dx = 0.

Learn how to find the linear function f(x) that satisfies the conditions \int_{-1}^1 f(x) dx = -1 and \int_0^1 x f(x) dx = 0. This problem uses calculus concepts like definite integrals and linear functions. Suitable for college-level calculus students.

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