Find the domain of the function. \[ f(x)=\sqrt{4 x-36} \] The domain is \( \square \) : (Type your answer in interval notation.)

To find the domain of the function \( f(x)=\sqrt{4x-36} \), we need to ensure that the expression under the square root is non-negative. This leads us to solve the inequality: \[ 4x - 36 \geq 0. \] By rearranging, we find: \[ 4x \geq 36 \] Dividing both sides by 4: \[ x \geq 9. \] Thus, the domain of the function is all values of \( x \) that are greater than or equal to 9. In interval notation, this can be expressed as \( [9, \infty) \). To visualize this, imagine a number line: the domain starts at 9 and continues forever to the right, covering all numbers that make the function valid. In real-world applications, functions with square roots are often seen in physics, particularly when calculating distances or energy changes, where only non-negative values make sense. Here, the function models scenarios like calculating the potential energy of an object based on height, where \( x \) represents some form of height or distance that must not be less than 9 units!