The image contains the following problem:

Let $f$ and $g$ be sets of real numbers defined as follows:

$f = \{x \mid x \leq 6\}$

$g = \{x \mid x \geq 8\}$

We need to find:

1. $f \cap g$ (the intersection of $f$ and $g$)
2. $f \cup g$ (the union of $f$ and $g$)

If the set is empty, we should write $\emptyset$.

Solutions:

1. Intersection ($f \cap g$):

   • $f \cap g$ represents the common elements between sets $f$ and $g$.
   • From the definitions, $f = \{x \mid x \leq 6\}$ and $g = \{x \mid x \geq 8\}$.
   • There are no elements that satisfy both conditions $x \leq 6$ and $x \geq 8$ simultaneously.
   • Therefore, $f \cap g = \emptyset$.

2. Union ($f \cup g$):

   • $f \cup g$ represents all elements that are in either set $f$ or set $g$.
   • For $f = \{x \mid x \leq 6\}$ and $g = \{x \mid x \geq 8\}$, the union combines all values less than or equal to 6 with all values greater than or equal to 8.
   • Therefore, $f \cup g = \{x \mid x \leq 6 \text{ or } x \geq 8\}$.

Final Answers:

1. $f \cap g = \emptyset$
2. $f \cup g = \{x \mid x \leq 6 \text{ or } x \geq 8\}$

Would you like any further details or have any questions?

Here are some related questions:

1. How do we find the intersection and union of more complex sets?
2. What are some real-life applications of set intersections and unions?
3. How would the result change if $f$ and $g$ had overlapping values?
4. Can we represent these sets and their operations visually using a number line or Venn diagrams?
5. What are the properties of sets and operations like intersection and union?

Tip: When working with set operations, always consider visualizing the problem using Venn diagrams or a number line for better understanding!