Let f=​{(- 4,2​),(-1, 4)(3,0)} and g={(-3,3),(1,3)(2,0)(4,-1)}. Find f(g(x))

To find \( f(g(x)) \), also known as the composition of functions \( f \circ g \), we substitute the outputs of \( g(x) \) into the function \( f(x) \).

Given the sets of points for \( f \) and \( g \):

- \( f = \{(-4, 2), (-1, 4), (3, 0)\} \)
- \( g = \{(-3, 3), (1, 3), (2, 0), (4, -1)\} \)

We need to follow these steps:
1. Identify the value of \( g(x) \) for each \( x \) in the domain of \( g \).
2. Substitute these \( g(x) \) values into \( f(x) \), checking if \( f \) is defined for those values.

### Step 1: Find \( g(x) \)
- For \( x = -3 \), \( g(-3) = 3 \).
- For \( x = 1 \), \( g(1) = 3 \).
- For \( x = 2 \), \( g(2) = 0 \).
- For \( x = 4 \), \( g(4) = -1 \).

### Step 2: Find \( f(g(x)) \)
Now, we substitute the values of \( g(x) \) into \( f(x) \):

- For \( g(-3) = 3 \), we find \( f(3) \). From the definition of \( f \), \( f(3) = 0 \), so \( f(g(-3)) = 0 \).
- For \( g(1) = 3 \), we find \( f(3) \) again. Thus, \( f(g(1)) = 0 \).
- For \( g(2) = 0 \), we check if \( f(0) \) is defined, but \( 0 \) is not in the domain of \( f \), so \( f(g(2)) \) is **undefined**.
- For \( g(4) = -1 \), we find \( f(-1) \). From the definition of \( f \), \( f(-1) = 4 \), so \( f(g(4)) = 4 \).

### Conclusion:
The composition \( f(g(x)) \) is:
- \( f(g(-3)) = 0 \)
- \( f(g(1)) = 0 \)
- \( f(g(2)) = \text{undefined} \)
- \( f(g(4)) = 4 \)

Would you like further details on function composition?

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**Relative questions**:
1. How do you determine the domain of a composite function?
2. What happens when a value in \( g(x) \) is not in the domain of \( f(x) \)?
3. Can you graph \( f \circ g \) to visualize its behavior?
4. How would \( f \circ g \) change if a new point were added to either function?
5. What is the significance of an undefined value in function composition?

**Tip**: Always check if the output of \( g(x) \) lies within the domain of \( f(x) \) before evaluating the composite function.

Learn how to find the composition of functions f(g(x)) with f and g defined by sets of points. This problem covers function composition, domain restrictions, and undefined values. Suitable for students in Grades 10-12.

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