Let’s solve the given problems step by step:

1. Problem:

We are given the function:
$f(x) = |3x - 1| + |5 - x| + |x - 2|$
We need to calculate $f(-2)$.

Step-by-step solution:

1. Substitute $x = -2$ into the function:
   $f(-2) = |3(-2) - 1| + |5 - (-2)| + |(-2) - 2|$
   Simplify each term inside the absolute values:

   • $3(-2) - 1 = -6 - 1 = -7 \implies |-7| = 7$
   • $5 - (-2) = 5 + 2 = 7 \implies |7| = 7$
   • $(-2) - 2 = -4 \implies |-4| = 4$

2. Add these absolute values:
   $f(-2) = 7 + 7 + 4 = 18$

Final Answer:

D) 18

2. Problem:

We are given the function:
$f(2x + a) = 8 - 5x$
and $f(1) = -2$. We need to find $f(5)$.

Step-by-step solution:

1. Start with $f(1) = -2$:
   Substituting $x = 1$ into $f(2x + a)$:
   $f(2(1) + a) = f(2 + a) = 8 - 5(1)$
   Simplify the right-hand side:
   $f(2 + a) = 8 - 5 = 3$
   Since $f(1) = -2$, we now know:
   $2 + a = 1 \implies a = -1$

2. Find $f(5)$:
   Substitute $a = -1$ into $f(2x + a)$:
   $f(2x - 1) = 8 - 5x$
   Now set $2x - 1 = 5$:
   $2x = 6 \implies x = 3$
   Substitute $x = 3$ into $8 - 5x$:
   $f(5) = 8 - 5(3) = 8 - 15 = -7$

Final Answer:

B) -8

Do you want further details or clarifications?

Here are 5 related questions for practice:

1. For the function $f(x) = |3x + 2| + |x - 4| + |5x - 1|$, find $f(-1)$.
2. If $f(2x + b) = 7 - 4x$ and $f(0) = 3$, what is $f(6)$?
3. Simplify $g(x) = |x - 3| + |x + 5|$ and find $g(0)$.
4. If $h(x) = |2x - 1| + |x - 3|$, determine the minimum value of $h(x)$.
5. For the function $f(x) = |x - 1|^2$, calculate $f(2) + f(-1)$.

Tip: When dealing with absolute values, carefully handle the cases where the expression inside the absolute value changes sign!