To perform the operation $(f \cdot g)(x)$, which represents the composition of two functions, we evaluate $g(x)$ first and then substitute that result into $f(x)$. This is different from the product of the functions $f(x) \times g(x)$. Here’s how you can calculate $(f \cdot g)(x)$ for each of the given pairs of functions:

1. $f(x) = x + 1$, $g(x) = 2x - 3$

$(f \cdot g)(x) = f(g(x)) = f(2x - 3)$ Substitute $2x - 3$ into $f(x)$: $f(2x - 3) = (2x - 3) + 1 = 2x - 2$ So, $(f \cdot g)(x) = 2x - 2$.

3. $f(x) = x^2 + 1$, $g(x) = x - 3$

$(f \cdot g)(x) = f(g(x)) = f(x - 3)$ Substitute $x - 3$ into $f(x)$: $f(x - 3) = (x - 3)^2 + 1$ Expand $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$ So, $f(x - 3) = x^2 - 6x + 9 + 1 = x^2 - 6x + 10$ Thus, $(f \cdot g)(x) = x^2 - 6x + 10$.

5. $f(x) = x - 2$, $g(x) = \frac{1}{x}$

$(f \cdot g)(x) = f\left(\frac{1}{x}\right)$ Substitute $\frac{1}{x}$ into $f(x)$: $f\left(\frac{1}{x}\right) = \frac{1}{x} - 2$ So, $(f \cdot g)(x) = \frac{1}{x} - 2$.

7. $f(x) = 3x^2 + 1$, $g(x) = x - 3$

$(f \cdot g)(x) = f(g(x)) = f(x - 3)$ Substitute $x - 3$ into $f(x)$: $f(x - 3) = 3(x - 3)^2 + 1$ Expand $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$ So, $f(x - 3) = 3(x^2 - 6x + 9) + 1 = 3x^2 - 18x + 27 + 1 = 3x^2 - 18x + 28$ Thus, $(f \cdot g)(x) = 3x^2 - 18x + 28$.

9. $f(x) = \frac{2}{x}$, $g(x) = \frac{1}{x}$

$(f \cdot g)(x) = f\left(\frac{1}{x}\right)$ Substitute $\frac{1}{x}$ into $f(x)$: $f\left(\frac{1}{x}\right) = \frac{2}{\frac{1}{x}} = 2x$ So, $(f \cdot g)(x) = 2x$.

Summary of Answers:

1. $(f \cdot g)(x) = 2x - 2$
2. $(f \cdot g)(x) = x^2 - 6x + 10$
3. $(f \cdot g)(x) = \frac{1}{x} - 2$
4. $(f \cdot g)(x) = 3x^2 - 18x + 28$
5. $(f \cdot g)(x) = 2x$

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Further Questions:

1. How do you differentiate between function composition and multiplication?
2. What is the difference between $(f \cdot g)(x)$ and $(g \cdot f)(x)$?
3. How do you expand a binomial like $(x - 3)^2$?
4. What happens if $g(x)$ is a constant function?
5. Can function composition be applied to trigonometric functions?
6. What is the significance of the order in function composition?
7. How do you interpret the result of $(f \cdot g)(x)$ graphically?
8. How can you verify if a function is invertible before composition?

Tip: When composing functions, always ensure that the output of the inner function (g(x)) falls within the domain of the outer function (f(x)).