Use graphing technology to find the range of the function f, of, x, equals, square root of, x, plus, 5, end square root, minus, 4, .f(x)=

x+5

−4.

1 answer

To find the range of the function \( f(x) = \sqrt{x + 5} - 4 \), we first analyze the components of the function:

  1. Identify the domain: The function includes a square root, which means the expression inside the radical must be non-negative: \[ x + 5 \geq 0 \implies x \geq -5. \] Therefore, the domain of \( f(x) \) is \( x \in [-5, \infty) \).

  2. Behavior of the function:

    • When \( x = -5 \): \[ f(-5) = \sqrt{-5 + 5} - 4 = \sqrt{0} - 4 = -4. \]
    • As \( x \) increases beyond -5, \( \sqrt{x + 5} \) also increases. Specifically, as \( x \to \infty \): \[ f(x) \to \infty \text{ (since } \sqrt{x + 5} \text{ will approach infinity)}. \]
  3. Combining the above observations:

    • The minimum value of \( f(x) \) occurs at \( x = -5 \), which is \( -4 \).
    • As \( x \) increases, \( f(x) \) increases without bound.

Thus, the range of the function is: \[ \text{Range of } f(x) = [-4, \infty). \]

In summary:

  • The minimum value of \( f(x) \) is -4, which occurs at the left endpoint of the domain.
  • There is no upper bound on \( f(x) \) as \( x \) increases.
  • Therefore, the final answer is: \[ \text{Range} = [-4, \infty). \]