Let's solve each function for the given values step by step:

1. Find $f(1)$: Function: $f(x) = -2|x - 5| + 2$

   Substituting $x = 1$: $f(1) = -2|1 - 5| + 2 = -2| -4 | + 2 = -2(4) + 2 = -8 + 2 = -6$

   Result: $f(1) = -6$

2. Find $f(4)$: Function: $f(x) = x^2 - 10x + 23$

   Substituting $x = 4$: $f(4) = (4)^2 - 10(4) + 23 = 16 - 40 + 23 = -24 + 23 = -1$

   Result: $f(4) = -1$

3. Find $f(-3)$: Function: $f(x) = -x^2 + 1$

   Substituting $x = -3$: $f(-3) = -(-3)^2 + 1 = -(9) + 1 = -9 + 1 = -8$

   Result: $f(-3) = -8$

4. Find $f(8)$: Function: $f(x) = |4x - 1| - 14$

   Substituting $x = 8$: $f(8) = |4(8) - 1| - 14 = |32 - 1| - 14 = |31| - 14 = 31 - 14 = 17$

   Result: $f(8) = 17$

5. Find $f(-2)$ (for two different functions):

   • First Function: $f(x) = -x^3 + 5$

     Substituting $x = -2$: $f(-2) = -(-2)^3 + 5 = -( -8 ) + 5 = 8 + 5 = 13$

     Result: $f(-2) = 13$

   • Second Function: $f(x) = x^2 + 4x - 1$

     Substituting $x = -2$: $f(-2) = (-2)^2 + 4(-2) - 1 = 4 - 8 - 1 = -4 - 1 = -5$

     Result: $f(-2) = -5$

Final Answers:

1. $f(1) = -6$
2. $f(4) = -1$
3. $f(-3) = -8$
4. $f(8) = 17$
5. $f(-2) = 13$ (first function), $f(-2) = -5$ (second function)

Would you like a breakdown of any particular step?

Related Questions:

1. What happens to the value of $f(x)$ as $x$ approaches a large positive number in $f(x) = |4x - 1| - 14$?
2. How does the graph of $f(x) = -x^3 + 5$ behave for large negative values of $x$?
3. What is the vertex of the parabola $f(x) = x^2 - 10x + 23$?
4. How does the absolute value impact the function $f(x) = -2|x - 5| + 2$?
5. What is the domain and range of $f(x) = -x^2 + 1$?

Tip:

Always double-check your substitutions, especially when dealing with absolute values and negative exponents, as small mistakes can lead to incorrect results.