To find the functional values of the given operations $\frac{f}{g}(x)$, $(fg)(x)$, $(f \circ g)(x)$, and $(g \circ f)(x)$, we must understand the steps to compute these operations for given functions $f(x)$ and $g(x)$.

Let’s define $f(x)$ and $g(x)$ to work with specific examples:

Example Functions:

• $f(x) = 2x + 3$
• $g(x) = x^2 - 1$

1. Division of functions: $\frac{f}{g}(x)$:

This operation involves dividing $f(x)$ by $g(x)$: $\frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{2x + 3}{x^2 - 1}.$

• Compute $f(x) = 2x + 3$.
• Compute $g(x) = x^2 - 1$.
• Combine them as a fraction: $\frac{f}{g}(x) = \frac{2x + 3}{x^2 - 1}.$

Domain Restriction: Since division by zero is undefined, we exclude values of $x$ where $g(x) = 0$. Solve $x^2 - 1 = 0$: $x = \pm 1.$ Therefore, the domain of $\frac{f}{g}(x)$ is all real numbers except $x = \pm 1$.

2. Product of functions: $(fg)(x)$:

This operation involves multiplying $f(x)$ and $g(x)$: $(fg)(x) = f(x) \cdot g(x) = (2x + 3)(x^2 - 1).$

• Expand the expression: $(fg)(x) = (2x + 3)(x^2 - 1) = 2x^3 - 2x + 3x^2 - 3.$
• Combine like terms: $(fg)(x) = 2x^3 + 3x^2 - 2x - 3.$

3. Composition of functions: $(f \circ g)(x)$:

This operation involves substituting $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x)) = f(x^2 - 1).$

• Start with $f(x) = 2x + 3$.
• Replace $x$ in $f(x)$ with $g(x) = x^2 - 1$: $f(g(x)) = 2(x^2 - 1) + 3.$
• Simplify: $f(g(x)) = 2x^2 - 2 + 3 = 2x^2 + 1.$

4. Composition of functions: $(g \circ f)(x)$:

This operation involves substituting $f(x)$ into $g(x)$: $(g \circ f)(x) = g(f(x)) = g(2x + 3).$

• Start with $g(x) = x^2 - 1$.
• Replace $x$ in $g(x)$ with $f(x) = 2x + 3$: $g(f(x)) = (2x + 3)^2 - 1.$
• Expand $(2x + 3)^2$: $(2x + 3)^2 = 4x^2 + 12x + 9.$
• Subtract 1: $g(f(x)) = 4x^2 + 12x + 9 - 1 = 4x^2 + 12x + 8.$

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Final Results:

1. $\frac{f}{g}(x) = \frac{2x + 3}{x^2 - 1}, \; x \neq \pm 1$.
2. $(fg)(x) = 2x^3 + 3x^2 - 2x - 3$.
3. $(f \circ g)(x) = 2x^2 + 1$.
4. $(g \circ f)(x) = 4x^2 + 12x + 8$.

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Would you like a graph of these functions or further clarifications?

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Related Questions:

1. What are the domains of $f(x)$, $g(x)$, and the resulting compositions?
2. How would these operations differ if $f(x)$ or $g(x)$ had discontinuities?
3. Can you explain the difference between $f \circ g(x)$ and $g \circ f(x)$?
4. What happens when $g(x)$ is a trigonometric or exponential function?
5. How do we interpret these operations in terms of real-world applications?

Tip:

Always check the domains of the individual functions and ensure valid substitutions or operations when dealing with composition or division of functions.