To construct a function that satisfies the given conditions, we can use a piecewise function approach.
1. For the first question, let's start by considering the conditions stated:
- The domain is all real numbers, so we don't have any restrictions on x.
- The function has no maximum and no minimum on the interval [1,3], meaning it cannot reach the extremes within that range.
- We need to satisfy f(1) = 1 and f(3) = -1, but there shouldn't exist a value c between 1 and 3 such that f(x) = 0.
To meet these conditions, we can construct a piecewise function as follows:
```
{ x - 1 if x < 1 or x > 3
f(x) = {
{ 2(3 - x) - 1 if 1 <= x <= 3
```
Explanation of the function:
- For x < 1 or x > 3, we define f(x) as x - 1, allowing us to avoid any maximum or minimum within the interval [1,3].
- For 1 ≤ x ≤ 3, we define f(x) as 2(3 - x) - 1. This equation satisfies f(1) = 1 and f(3) = -1. Additionally, it ensures that f(x) ≠ 0 for any value of x within the interval [1,3].
2. For the second question, the conditions are as follows:
- The function f(x) should be continuous for all x, meaning there are no abrupt jumps or breaks in the graph.
- The range of f(x) should be between -10 and 10, inclusively.
- The function does not have a maximum or minimum on the interval [negative infinity, infinity].
To satisfy these conditions, we can use a function that oscillates infinitely between -10 and 10, such as a sine or cosine function. For example:
```
f(x) = 10 * sin(x)
```
Explanation of the function:
- By using the sine function, f(x) = 10 * sin(x), we ensure that the function oscillates symmetrically about the x-axis, causing it to have neither a maximum nor a minimum on the interval [negative infinity, infinity].
- Since the range of the sine function is between -1 and 1, we scale it by 10 to satisfy the condition that -10 ≤ f(x) ≤ 10 for all x.
Note: These are just two possible ways to construct functions that satisfy the given conditions. There are many other possible functions that meet the requirements.