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When dealing with functions such as \( (f+g)(x) = \sqrt{2x - 7} + \sqrt{4x + 15} \), it is crucial to understand when each component is defined. This means looking at their individual domains.

The domain of a function is the set of input values \( x \) for which the function is defined. For the function \( f(x) = \sqrt{2x - 7} \), the expression under the square root, \( 2x - 7 \), must be non-negative. This results in the condition \( x \geq \frac{7}{2} \).

Similarly, for \( g(x) = \sqrt{4x + 15} \), the condition comes from \( 4x + 15 \geq 0 \), leading to \( x \geq -\frac{15}{4} \).

To find the domain of the combined function \( (f+g)(x) \), you need to take the intersection of the domains of \( f(x) \) and \( g(x) \). This means looking for the largest interval where both functions are defined. Since \( x \geq \frac{7}{2} \) is more restrictive than \( x \geq -\frac{15}{4} \), the domain of \( (f+g)(x) \) is \( x \geq \frac{7}{2} \).

Understanding square root functions is essential in determining their domains. A square root function such as \( f(x) = \sqrt{2x - 7} \) is only defined when the number inside the square root, known as the radicand, is zero or positive.

Mathematically, this is expressed as \( 2x - 7 \geq 0 \). Solving for \( x \), gives the minimum value \( x = \frac{7}{2} \). So, \( f(x) \) is only defined for \( x \geq \frac{7}{2} \).

• Square roots are non-negative, meaning they produce only positive or zero outputs.
• Under the square root symbol, the expression must not be negative to be real.

Likewise, for \( g(x) = \sqrt{4x + 15} \), the condition \( 4x + 15 \geq 0 \) ensures the function remains defined. The solution \( x \geq -\frac{15}{4} \) determines its domain. These checks ensure that both functions are working within their valid input ranges.

Analytical confirmation helps ensure that the domains determined are correct. Once the arithmetic and logic have been processed to find the intersection of domains, an analytical walkthrough can verify the solution.

Starting from each function, check:

• For \( f(x) = \sqrt{2x - 7} \), the defined range is \( x \geq \frac{7}{2} \).
• For \( g(x) = \sqrt{4x + 15} \), the defined range is \( x \geq -\frac{15}{4} \).

To confirm the domain of \( (f+g)(x) = \sqrt{2x - 7} + \sqrt{4x + 15} \), analyze both conditions. The intersection, where both these conditions hold true, requires the stricter domain of \( \frac{7}{2} \geq x \).

Using graphing tools can be helpful for visual learners. A graph provides a visual overlap, confirming that the shared region begins at \( x = \frac{7}{2} \). Thus, both the analytical and graphical checks align, confirming the function's combined domain.