first handling the defintion given above the exercise, we that "f" is a funktion 
f: A implies B;
 and that 
                          "S subset B"

. and that we defined the inverse image of S to be the subset of A, whose elements are precisely all preimages of all elements of S we denote the iverse image of S by 
                            "1/f(S)"

so 
                "f^(-1)(S)={a in A | f(a) in S}"

let 
                              "f"

 be the function from R to R defined by 
                          "f(x) = x^2"

find:                           "1/f({1})"


 f^(-1)({x| 0<x<1}) 0.5;
f^(-1)({x | x>4})  5;




To solve the problem, we need to handle inverse images of sets under the function \( f(x) = x^2 \) where \( f: \mathbb{R} \to \mathbb{R} \).

### 1. **Understanding the inverse image of a set:**
The inverse image \( f^{-1}(S) \) of a subset \( S \subseteq \mathbb{R} \) is defined as:
\[
f^{-1}(S) = \{a \in A \mid f(a) \in S\}.
\]
In this case, \( f(x) = x^2 \), so we're looking for all values of \( x \in \mathbb{R} \) such that \( x^2 \in S \).

Now, let's find the inverse images of the given sets:

### 2. **Finding \( f^{-1}(\{1\}) \):**

We are asked to find \( f^{-1}(\{1\}) \), meaning we need to find all \( x \in \mathbb{R} \) such that:
\[
f(x) = x^2 = 1.
\]
This gives two possible solutions:
\[
x^2 = 1 \quad \Rightarrow \quad x = \pm 1.
\]
Thus, \( f^{-1}(\{1\}) = \{-1, 1\} \).

### 3. **Finding \( f^{-1}(\{x \mid 0 < x < 1\}) \):**

Next, we need to find \( f^{-1}(\{x \mid 0 < x < 1\}) \), which means finding all \( x \in \mathbb{R} \) such that:
\[
0 < x^2 < 1.
\]
The solutions to this inequality are:
\[
-1 < x < 1.
\]
Thus, \( f^{-1}(\{x \mid 0 < x < 1\}) = (-1, 1) \).

### 4. **Finding \( f^{-1}(\{x \mid x > 4\}) \):**

Finally, we need to find \( f^{-1}(\{x \mid x > 4\}) \), which means finding all \( x \in \mathbb{R} \) such that:
\[
x^2 > 4.
\]
This inequality holds when:
\[
x > 2 \quad \text{or} \quad x < -2.
\]
Thus, \( f^{-1}(\{x \mid x > 4\}) = (-\infty, -2) \cup (2, \infty) \).

### Final Answers:
1. \( f^{-1}(\{1\}) = \{-1, 1\} \)
2. \( f^{-1}(\{x \mid 0 < x < 1\}) = (-1, 1) \)
3. \( f^{-1}(\{x \mid x > 4\}) = (-\infty, -2) \cup (2, \infty) \)

---

Do you want details or have any questions?

Here are 5 related questions to explore further:
1. What is the inverse image of a set for a different function, like \( f(x) = x^3 \)?
2. How does the inverse image change when the function involves absolute values, like \( f(x) = |x| \)?
3. Can we generalize the process of finding inverse images for any polynomial function?
4. What happens to the inverse image if we restrict the function \( f(x) = x^2 \) to positive real numbers only?
5. How does the concept of inverse images relate to preimages in topology?

**Tip:** Always remember that the inverse image of a set might contain more than one value, even if the original set is a single point. This happens especially with non-injective functions like \( f(x) = x^2 \).

The query submitted by the user:first handling the defintion given above the exercise, we that "f" is a funktion f: A implies B; and that "S subset B". and that we defined the inverse image of S to be the subset of A, whose elements are precisely all preimages of all elements of S we denote the iverse image of S by "1/f(S)" so "f^(-1)(S)={a in A | f(a) in S}" let "f" be the function from R to R defined by "f(x) = x^2" find: "1/f({1})" f^(-1)({x| 0<x<1}) 0.5; f^(-1)({x | x>4}) 5;

Learn how to calculate the inverse image of sets under the function f(x) = x^2, including detailed solutions for f^(-1)({1}), f^(-1)({x | 0 < x < 1}), and f^(-1)({x | x > 4}). This covers key concepts such as inverse functions, preimages, and set theory, suitable for advanced high school or college-level students.

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