To determine the function $f = g(x - 5) - 3$ as a set of ordered pairs, we'll follow these steps:

Step-by-Step Process:

1. Understand the transformation:

   • The function $f(x)$ is a transformation of $g(x)$.
   • $g(x-5)$ means that we take the input $x$, shift it by 5 to the right, and then evaluate it using the given points in $g$.
   • After calculating $g(x-5)$, we subtract 3 from the result to get $f(x)$.

2. Given points in $g$:

   • $g = \{(-1, -2), (0, 1), (5, 16), (7, 22), (10, 31)\}$.

   This means $g(-1) = -2$, $g(0) = 1$, $g(5) = 16$, $g(7) = 22$, and $g(10) = 31$.

3. Apply the transformation $f = g(x-5) - 3$:

   We need to find $f(x)$ for the points corresponding to these transformations:

   • For each point in $g$, we solve for $f(x)$ by finding $x$ such that $g(x-5) = g(y)$, where $y = x - 5$.

Now calculate the new points:

1. $f(4) = g(4-5) - 3 = g(-1) - 3 = -2 - 3 = -5$

   • So, one point is $(4, -5)$.

2. $f(5) = g(5-5) - 3 = g(0) - 3 = 1 - 3 = -2$

   • So, one point is $(5, -2)$.

3. $f(10) = g(10-5) - 3 = g(5) - 3 = 16 - 3 = 13$

   • So, one point is $(10, 13)$.

4. $f(12) = g(12-5) - 3 = g(7) - 3 = 22 - 3 = 19$

   • So, one point is $(12, 19)$.

5. $f(15) = g(15-5) - 3 = g(10) - 3 = 31 - 3 = 28$

   • So, one point is $(15, 28)$.

Final set of ordered pairs for $f$:

$f = \{(4, -5), (5, -2), (10, 13), (12, 19), (15, 28)\}$

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Would you like further details or explanations on any step?

Here are 5 related questions to expand your understanding:

1. How does shifting a function horizontally affect its graph?
2. How would the function change if the transformation was $g(x+5)$ instead of $g(x-5)$?
3. What happens when you add or subtract constants from a function's output?
4. How can you determine the inverse of a function?
5. What is the geometric interpretation of the transformation applied to $g(x)$?

Tip: Always remember, shifting $x$ horizontally is linked to the opposite direction of the sign inside the parentheses. $g(x-5)$ shifts to the right, and $g(x+5)$ shifts to the left.