The problem presented is:

Given $f(X) = X^n - 2$ and ( f(3)The problem in the image is:

Given $f(X) = X^n - 2$, and that $f(3) = 7$, determine the degree $n$ of the function $f(X)$.

Solution:

We have the equation:

$f(X) = X^n - 2$

Substitute $X = 3$ and use the fact that $f(3) = 7$:

$f(3) = 3^n - 2 = 7$

Solving for $n$:

$3^n - 2 = 7$

Add 2 to both sides:

$3^n = 9$

Now, since $3^2 = 9$, it follows that $n = 2$.

Thus, $f(X) = X^2 - 2$, and the function is of second degree.

Answer:

The degree of the function is 2.

Would you like more details or have any questions about this?

Here are 5 related questions to expand on this topic:

1. How would the degree change if the constant term in $f(X)$ was different?
2. What is the general method for determining the degree of a polynomial function?
3. If $f(X) = X^n + 5$, and $f(2) = 13$, how would you solve for $n$?
4. How does the degree of a polynomial affect its graph's shape?
5. What are some applications of polynomial degree in real-world problems?

Tip: The degree of a polynomial tells you the highest power of the variable and often determines the overall shape and behavior of its graph!