Math worksheet focused on evaluating rational functions, including simplification and substitution exercises.

I'll solve the problem in the image, which is about evaluating rational functions. Let me go through each part step by step.

Part 1: Evaluate each function at the specified value

Problem 1: f(x) = (x+1)/(x-3); x = -1
- Substitute x = -1 into the function: f(-1) = (-1+1)/(-1-3) = 0/(-4) = 0

Problem 2: f(x) = (x²-2x)/(x-2); x = 4
- Substitute x = 4 into the function: f(4) = (4²-2×4)/(4-2) = (16-8)/2 = 8/2 = 4

Problem 3: f(x) = (5x-3)/(2x+1); x = 2
- Substitute x = 2 into the function: f(2) = (5×2-3)/(2×2+1) = (10-3)/(4+1) = 7/5

Problem 4: f(x) = (3x+1)/(x-4); x = 4
- Substitute x = 4 into the function: f(4) = (3×4+1)/(4-4) = (12+1)/0 = 13/0
- This is undefined because we cannot divide by zero.

Part 2: Evaluate each function

Problem 5: f(x) = (x²-4)/(x-2); for x ≠ 2
- We can factor the numerator: x²-4 = (x-2)(x+2)
- So f(x) = ((x-2)(x+2))/(x-2)
- For x ≠ 2, we can cancel (x-2) from numerator and denominator: f(x) = x+2
- So for any x ≠ 2, f(x) = x+2

Problem 6: f(x) = (8x)/(x-1); for x ≠ 1
- This function is already simplified.
- For any x ≠ 1, f(x) = 8x/(x-1)

Problem 7: f(x) = 1/(x+2); for x ≠ -2
- This function is already simplified.
- For any x ≠ -2, f(x) = 1/(x+2)

Problem 8: f(x) = (x+3)/(x(x-1)); for x ≠ 0, 1
- This function is already simplified.
- For any x ≠ 0, 1, f(x) = (x+3)/(x(x-1))

Part 3: What is the value of x that makes f(x) = (x²-4)/(x-2) undefined?

The function f(x) = (x²-4)/(x-2) is undefined when the denominator equals zero.
- Set the denominator equal to zero: x-2 = 0
- Solve for x: x = 2

Therefore, the function is undefined when x = 2.

Looking at the multiple choice options:
a) -∞
b) 0
c) 2
d) -2

The correct answer is c) 2.

This matches with what we found in Problem 5, where we noted that x ≠ 2 because the function would be undefined at that point.