Range of a Radical Function
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The range of which function includes $-4$?
A) $y = \sqrt{x} + 5$
B) $y = \sqrt{x - 5}$
C) $y = \sqrt{x} - 5$
D) $y = \sqrt{x + 5}$
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Step by Step Written Solution
Hi Aiden, let's figure out which of these radical functions has a range that includes negative four.
Range of Radical Functions
First, recall that for a basic square root function, y equals the square root of x, the output is always zero or greater.
Let's analyze the range for each of the given options by looking at the vertical shifts.
Analyzing Options
| Function | Range |
|---|---|
| y = \sqrt{x} + 5 | y \ge 5 |
| y = \sqrt{x - 5} | y \ge 0 |
| y = \sqrt{x} - 5 | y \ge -5 |
| y = \sqrt{x + 5} | y \ge 0 |
For the first option, y equals the square root of x plus five, the entire graph is shifted up by five units. So the range is y is greater than or equal to five. This does not include negative four.
The second option has a horizontal shift, but no vertical shift. So the range remains y is greater than or equal to zero. This also does not include negative four.
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