To analyze the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = \sqrt{x + 1} - 3 \), let's look into their properties systematically:
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Domains: Both functions involve the square root \( \sqrt{x + 1} \). The expression inside the square root must be non-negative for the function to be defined.
- For \( f(x) \), we have \( x + 1 \geq 0 \), thus \( x \geq -1 \).
- For \( g(x) \), the same inequality holds: \( x + 1 \geq 0 \) implies \( x \geq -1 \).
- Therefore, both functions have the same domain: \( [-1, \infty) \).
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X-intercepts: To find the x-intercepts, we set \( y = 0 \).
- For \( f(x) = 0 \): \[ \sqrt{x + 1} - 2 = 0 \quad \Rightarrow \quad \sqrt{x + 1} = 2 \quad \Rightarrow \quad x + 1 = 4 \quad \Rightarrow \quad x = 3. \]
- For \( g(x) = 0 \): \[ \sqrt{x + 1} - 3 = 0 \quad \Rightarrow \quad \sqrt{x + 1} = 3 \quad \Rightarrow \quad x + 1 = 9 \quad \Rightarrow \quad x = 8. \]
- Therefore, the x-intercepts are \( (3, 0) \) for \( f(x) \) and \( (8, 0) \) for \( g(x) \), which are not the same.
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Y-intercepts: To find the y-intercepts, we evaluate both functions at \( x = -1 \):
- For \( f(-1) \): \[ f(-1) = \sqrt{-1 + 1} - 2 = \sqrt{0} - 2 = -2. \]
- For \( g(-1) \): \[ g(-1) = \sqrt{-1 + 1} - 3 = \sqrt{0} - 3 = -3. \]
- Therefore, the y-intercepts are \( (0, -2) \) for \( f(x) \) and \( (0, -3) \) for \( g(x) \), which are also not the same.
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Ranges: Looking at the ranges:
- The function \( f(x) = \sqrt{x + 1} - 2 \) achieves values starting from \( -2 \) (when \( x = -1 \), the lowest point in the domain) and goes to \( +\infty \) as \( x \) increases.
- The function \( g(x) = \sqrt{x + 1} - 3 \) starts at \( -3 \) (when \( x = -1\)) and also goes to \( +\infty \) for larger \( x \).
- Hence, the range for \( f(x) \) is \( [-2, \infty) \) and for \( g(x) \) is \( [-3, \infty) \), which are not the same.
Conclusion:
Comparing all the analysis above, the only statement that is true is:
The graphs have the same domains.