The given function is $f(x) = x^2 - 2$ with a specified domain $\{0, 4, 5, 7, 8\}$. We are tasked with finding the range of this function.

Step-by-step process:

1. Evaluate the function for each value in the domain:

   • For $x = 0$: $f(0) = 0^2 - 2 = -2$
   • For $x = 4$: $f(4) = 4^2 - 2 = 16 - 2 = 14$
   • For $x = 5$: $f(5) = 5^2 - 2 = 25 - 2 = 23$
   • For $x = 7$: $f(7) = 7^2 - 2 = 49 - 2 = 47$
   • For $x = 8$: $f(8) = 8^2 - 2 = 64 - 2 = 62$

2. List the outputs (range values): The range is the set of all the output values, which are: $\{-2, 14, 23, 47, 62\}$

Conclusion:

The range of the function $f(x) = x^2 - 2$ with the domain $\{0, 4, 5, 7, 8\}$ is $\{-2, 14, 23, 47, 62\}$.

Would you like further details or clarifications?

Here are some additional questions you might find useful:

1. What happens to the range if the domain includes negative numbers?
2. How does the range of a quadratic function generally behave?
3. What would the range be if the domain were all real numbers?
4. How would the function change if we added a constant to it?
5. What effect does squaring a number have on the range of the function?

Tip: When dealing with quadratic functions, always consider that their outputs are symmetric about their vertex, and that this symmetry affects the range.