Let's analyze and solve the given problems based on the image.

Problem 12: Evaluate $f(x) = -2 |x - 5|$ at each value and simplify.

a. $f(4)$

You already started solving this:

• $f(4) = -2 |4 - 5|$
• $= -2 |-1|$
• The absolute value of $-1$ is $1$, so $f(4) = -2 \times 1 = -2$.

Thus, the value of $f(4)$ is -2.

b. $f(x^2 + 5)$

Here, we evaluate the same function with $x^2 + 5$ inside the absolute value:

• $f(x^2 + 5) = -2 |x^2 + 5 - 5|$
• Simplifying inside the absolute value: $f(x^2 + 5) = -2 |x^2|$
• Since $|x^2| = x^2$ (because squaring a number always gives a non-negative result), we have $f(x^2 + 5) = -2 x^2$.

Thus, the simplified expression for $f(x^2 + 5)$ is -2x².

Would you like further clarification on any part of this?

Here are 5 questions related to this topic:

1. What happens if we evaluate $f(x)$ for $x > 5$?
2. How would the graph of $f(x) = -2 |x - 5|$ look like?
3. How does the value of $x$ affect the absolute value expression in $f(x) = -2 |x - 5|$?
4. What would $f(10)$ be, using the same function $f(x) = -2 |x - 5|$?
5. How would the result differ if the coefficient outside the absolute value were positive instead of negative?

Tip: Always simplify the expression inside absolute values before applying any coefficients outside.