Let's define a function [tex]\( f(x) \)[/tex] and find a number [tex]\( x_0 \)[/tex] such that [tex]\( f(x_0) = -3 \)[/tex]. The function is required to be neither a constant function nor the identity function [tex]\( f(x) = x \)[/tex].
### Step-by-Step Solution
1. Define the Function [tex]\( f(x) \)[/tex]:
Let's choose a linear function for [tex]\( f(x) \)[/tex] that meets the criteria. We can define [tex]\( f(x) \)[/tex] as:
[tex]\[
f(x) = 2x - 5
\][/tex]
2. Find [tex]\( x_0 \)[/tex] Such That [tex]\( f(x_0) = -3 \)[/tex]:
We want to find [tex]\( x_0 \)[/tex] such that:
[tex]\[
f(x_0) = -3
\][/tex]
Substitute [tex]\( f(x) = 2x - 5 \)[/tex] into the equation:
[tex]\[
2x_0 - 5 = -3
\][/tex]
3. Solve for [tex]\( x_0 \)[/tex]:
To find [tex]\( x_0 \)[/tex], solve the equation:
[tex]\[
2x_0 - 5 = -3
\][/tex]
Add 5 to both sides of the equation:
[tex]\[
2x_0 = -3 + 5
\][/tex]
Simplify the right side:
[tex]\[
2x_0 = 2
\][/tex]
Divide both sides by 2:
[tex]\[
x_0 = 1
\][/tex]
4. Verify the Solution:
Substitute [tex]\( x_0 = 1 \)[/tex] back into the function to verify:
[tex]\[
f(1) = 2(1) - 5 = 2 - 5 = -3
\][/tex]
This confirms that [tex]\( f(1) = -3 \)[/tex].
### Final Function and Value
- The function [tex]\( f(x) = 2x - 5 \)[/tex].
- The value [tex]\( x_0 = 1 \)[/tex].
[tex]\[
\boxed{f(x) = 2x - 5, \quad x_0 = 1, \quad y = -3}
\][/tex]
