The image contains mathematical problems related to piecewise functions and evaluating them under specific conditions. I'll analyze and explain the problems step by step.

1. Analyzing the piecewise function $f(x)$:

From the highlighted portion of the image:

• The function $f(x)$ is defined as: \begin{cases} x + 2 & \text{if } x \leq 3 \\ x^2 - 10 & \text{if } x > 3 \end{cases}$$

2. Evaluating $f(3)$:

• According to the definition, when $x \leq 3$, we use the equation $f(x) = x + 2$. $f(3) = 3 + 2 = 5$

3. Finding the value of $a$ when $f(a) = 5$:

• Case 1: If $a \leq 3$, use $f(x) = x + 2$: $f(a) = a + 2 = 5 \quad \Rightarrow \quad a = 3$

• Case 2: If $a > 3$, use $f(x) = x^2 - 10$: $f(a) = a^2 - 10 = 5 \quad \Rightarrow \quad a^2 = 15 \quad \Rightarrow \quad a = \pm\sqrt{15}.$ However, since $a > 3$, we only consider the positive root: $a = \sqrt{15}.$

So, $a = 3$ or $a = \sqrt{15}$.

4. Determining the correct answer:

The question highlights $a = 4$, which satisfies:

• Case 2: $f(4) = 4^2 - 10 = 16 - 10 = 6$, which is incorrect for $f(a) = 5$.

The accurate solution seems to indicate $a = 3$ or $a = \sqrt{15}$.

Would you like me to analyze another part of the image or explain further? Let me know!

Relative Questions:

1. What is the role of piecewise functions in real-life applications?
2. How can we derive solutions when piecewise functions have multiple domains?
3. Why is it important to consider constraints like $x > 3$ in piecewise functions?
4. What happens when $a$ is at the boundary (e.g., $x = 3$) in such functions?
5. How do we evaluate piecewise functions involving inequalities?

Tip:

Always double-check which part of a piecewise function applies to your input value to avoid mistakes!