The function $h$ is defined as $h(x)=\sqrt{3 x+7}$. For what value of $x$ does $h(x)=4$?
The problem asks for the value of \(x\) that satisfies the equation \(h(x) = 4\), where \(h(x) = \sqrt{3x + 7}\). We are given four options: A \(0\), B \(1\), C \(2\), and D \(3\). To solve this, we set the expression for \(h(x)\) equal to \(4\). The domain of the function is determined by the requirement that the expression inside the square root must be non-negative: \(3x + 7 \ge 0\), which simplifies to \(x \ge -\frac{7}{3}\). Any solution we find must satisfy this condition.
We start with the equation:
\[\sqrt{3x + 7} = 4\]
To isolate \(x\), we square both sides of the equation to eliminate the square root:
\[(\sqrt{3x + 7})^2 = 4^2\]
This simplifies to:
\[3x + 7 = 16\]
Now, we solve for \(x\) by first isolating the term with \(x\):
\[3x = 16 - 7\]
\[3x = 9\]
Next, divide by \(3\):
\[x = \frac{9}{3} = 3\]
The calculated value is \(x = 3\). We check if this value is in the domain of the function: since \(3 \ge -\frac{7}{3}\), the solution is valid. Checking our answer: \(h(3) = \sqrt{3(3) + 7} = \sqrt{9 + 7} = \sqrt{16} = 4\). This matches the given condition. Therefore, option D is the correct choice.
In real-life scenarios, functions like \(h(x) = \sqrt{3x + 7}\) often appear in physical modeling, such as determining the distance an object has traveled or the required size of a structural component based on certain inputs. For example, if an engineer is calculating the required width (\(x\)) of a support beam to withstand a certain load, they might encounter an equation like this. By solving for \(x\), they can ensure the structure is built correctly and safely. Understanding how to handle radical equations allows for solving these types of engineering and scientific problems accurately.