To find the range of the function \( f(x) = \sqrt{x + 5} - 4 \), we first analyze the components of the function:
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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 \( x \in [-5, \infty) \).
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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 \( x \to \infty \): \[ f(x) \to \infty \text{ (since } \sqrt{x + 5} \text{ will approach infinity)}. \]
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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: \[ \text{Range} = [-4, \infty). \]