Let us solve this step by step.

Problem Setup

We are given the following:

1. $f(x) = 3x^2 - 2$
2. $g(x) = x - 7$
3. $h(x) = \frac{x + 1}{x - 1}, \; x \neq 1$

We need to compute: $3f(2) - g(5) + h(3).$

Step-by-Step Solution

1. Compute $f(2)$: Using $f(x) = 3x^2 - 2$: $f(2) = 3(2^2) - 2 = 3(4) - 2 = 12 - 2 = 10.$

2. Compute $g(5)$: Using $g(x) = x - 7$: $g(5) = 5 - 7 = -2.$

3. Compute $h(3)$: Using $h(x) = \frac{x + 1}{x - 1}$: $h(3) = \frac{3 + 1}{3 - 1} = \frac{4}{2} = 2.$

4. Substitute into the given expression: $3f(2) - g(5) + h(3) = 3(10) - (-2) + 2.$

5. Simplify the expression: $3(10) = 30,$ $-(-2) = +2,$ $30 + 2 + 2 = 34.$

Final Answer:

$\boxed{34}$

Would you like further clarification or more examples related to this? Here are some possible related questions:

1. How is $f(x) = 3x^2 - 2$ derived, and what does it represent?
2. Could you verify $h(x)$ for other values of $x$ apart from $x = 3$?
3. How would the solution change if $h(x)$ had a different denominator?
4. What happens if $x = 1$ is used in $h(x)$, and why is it restricted?
5. How can similar questions be approached using graphing or visualization techniques?

Tip:

Always substitute values carefully in fraction-based functions to avoid dividing by zero.