This example focuses on finding the sum $f + g$, difference $f - g$, product $f \cdot g$, and quotient $\frac{f}{g}$ of two functions $f(x) = \sqrt{4x - 1}$ and $g(x) = \frac{1}{x}$, along with determining the domain of each resulting function. Let’s go over each part and how the domain is derived:

1. Sum $f + g$: $f + g = \sqrt{4x - 1} + \frac{1}{x}$

   • Domain: The domain of $f(x) = \sqrt{4x - 1}$ is $x \geq \frac{1}{4}$ because $4x - 1 \geq 0$ (ensuring the radicand is non-negative).
   • The domain of $g(x) = \frac{1}{x}$ is $x \neq 0$.
   • Combining these, the domain is $\left[\frac{1}{4}, \infty\right)$.

2. Difference $f - g$: $f - g = \sqrt{4x - 1} - \frac{1}{x}$

   • Domain: Similar reasoning as with $f + g$, so the domain remains $\left[\frac{1}{4}, \infty\right)$.

3. Product $f \cdot g$: $f \cdot g = \frac{\sqrt{4x - 1}}{x}$

   • Domain: Since both $f(x)$ and $g(x)$ must be defined, the domain is $\left[\frac{1}{4}, \infty\right)$.

4. Quotient $\frac{f}{g}$: $\frac{f}{g} = x \sqrt{4x - 1}$

   • Domain: We avoid division by zero from $g(x)$, resulting in the domain $\left[\frac{1}{4}, \infty\right)$.

Each operation maintains the initial restrictions on $x$ from both $f(x)$ and $g(x)$, ensuring the function outputs are real numbers.

Would you like more details on how these functions are combined or on the domain restrictions?

Related Questions

1. What would happen to the domains if $g(x)$ were a different function?
2. How would the process change if $f(x)$ involved a different root, like $\sqrt[3]{4x - 1}$?
3. What are some techniques for combining functions with complex expressions?
4. How would we find the range of each combined function in addition to the domain?
5. How do we interpret and visualize the domain on a number line or graph?

Tip:

When determining the domain of a combined function, always consider both functions' individual restrictions.