Questions: Consider the following function. f(x)=-2 sqrt(x)+5 Determine the domain and range of the original function. Express your answer in interval notation.

Solution Steps

To determine the domain and range of the function \( f(x) = -2 \sqrt{x} + 5 \), we need to consider the properties of the square root function and the transformations applied to it.

1. Domain: The square root function \( \sqrt{x} \) is defined for \( x \geq 0 \). Therefore, the domain of \( f(x) \) is \( x \geq 0 \).

2. Range: The function \( f(x) = -2 \sqrt{x} + 5 \) starts at \( f(0) = 5 \) and decreases as \( x \) increases because of the negative coefficient in front of the square root. The minimum value of \( f(x) \) occurs as \( x \) approaches infinity, which makes \( f(x) \) approach negative infinity. Therefore, the range of \( f(x) \) is \( (-\infty, 5] \).

The function \( f(x) = -2 \sqrt{x} + 5 \) involves a square root, which is defined for non-negative values of \( x \). Therefore, the domain of the function is given by: \[ \text{Domain: } [0, \infty) \]

To find the range, we evaluate the behavior of the function as \( x \) varies. At \( x = 0 \): \[ f(0) = -2 \sqrt{0} + 5 = 5 \] As \( x \) approaches infinity, \( \sqrt{x} \) increases without bound, leading \( f(x) \) to decrease towards negative infinity: \[ \lim_{x \to \infty} f(x) = -\infty \] Thus, the range of the function is: \[ \text{Range: } (-\infty, 5] \]

Final Answer