Question
Compare the graphs of f(x)=√(x+1)−2 and g(x)=∛(x+1)−2. Identify which of the following features is true about the graphs.
a. they have the same domains
b. they have the same y-intercept
c. the have the same ranges
d. they have the same x-intercept
a. they have the same domains
b. they have the same y-intercept
c. the have the same ranges
d. they have the same x-intercept
Answers
GPT-4o mini
To compare the graphs of \( f(x) = \sqrt{x+1} - 2 \) and \( g(x) = \sqrt[3]{x+1} - 2 \), let's analyze each feature one by one:
1. **Domains**:
- For \( f(x) \): The expression \( \sqrt{x+1} \) implies that \( x+1 \) must be greater than or equal to 0. Therefore, the domain of \( f(x) \) is \( x \geq -1 \), or in interval notation, \([-1, \infty)\).
- For \( g(x) \): The cube root function \( \sqrt[3]{x+1} \) is defined for all real numbers. So, the domain of \( g(x) \) is \((- \infty, \infty)\).
Conclusion: The domains are not the same.
2. **Y-Intercepts**:
- For \( f(x) \): The y-intercept occurs when \( x = 0 \).
\[
f(0) = \sqrt{0 + 1} - 2 = \sqrt{1} - 2 = 1 - 2 = -1
\]
- For \( g(x) \): The y-intercept occurs when \( x = 0 \).
\[
g(0) = \sqrt[3]{0 + 1} - 2 = \sqrt[3]{1} - 2 = 1 - 2 = -1
\]
Conclusion: Both functions have the same y-intercept at \(-1\).
3. **Ranges**:
- For \( f(x) \): Since \( \sqrt{x+1} \) starts from \(\sqrt{0}\) (which is \(0\)) when \(x = -1\), the minimum value of \( f(x) \) is \(0 - 2 = -2\). As \( x \) increases, \( f(x) \) increases without bound. Thus, the range of \( f(x) \) is \([-2, \infty)\).
- For \( g(x) \): The cube root function can produce all real values. Hence, \( g(x) \) can output values by shifting down by \(2\), so the range of \( g(x) \) is also all real numbers, or \((- \infty, \infty)\).
Conclusion: The ranges are not the same.
4. **X-Intercepts**:
- For \( f(x) \): The x-intercept occurs when \( f(x) = 0 \).
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3
\]
- For \( g(x) \): The x-intercept occurs when \( g(x) = 0 \).
\[
\sqrt[3]{x + 1} - 2 = 0 \implies \sqrt[3]{x + 1} = 2 \implies x + 1 = 8 \implies x = 7
\]
Conclusion: The x-intercepts are different: \( f(x) \) has an x-intercept at \( x = 3 \) and \( g(x) \) at \( x = 7 \).
Based on the analysis:
- **The true statement is**: **b. they have the same y-intercept.**
1. **Domains**:
- For \( f(x) \): The expression \( \sqrt{x+1} \) implies that \( x+1 \) must be greater than or equal to 0. Therefore, the domain of \( f(x) \) is \( x \geq -1 \), or in interval notation, \([-1, \infty)\).
- For \( g(x) \): The cube root function \( \sqrt[3]{x+1} \) is defined for all real numbers. So, the domain of \( g(x) \) is \((- \infty, \infty)\).
Conclusion: The domains are not the same.
2. **Y-Intercepts**:
- For \( f(x) \): The y-intercept occurs when \( x = 0 \).
\[
f(0) = \sqrt{0 + 1} - 2 = \sqrt{1} - 2 = 1 - 2 = -1
\]
- For \( g(x) \): The y-intercept occurs when \( x = 0 \).
\[
g(0) = \sqrt[3]{0 + 1} - 2 = \sqrt[3]{1} - 2 = 1 - 2 = -1
\]
Conclusion: Both functions have the same y-intercept at \(-1\).
3. **Ranges**:
- For \( f(x) \): Since \( \sqrt{x+1} \) starts from \(\sqrt{0}\) (which is \(0\)) when \(x = -1\), the minimum value of \( f(x) \) is \(0 - 2 = -2\). As \( x \) increases, \( f(x) \) increases without bound. Thus, the range of \( f(x) \) is \([-2, \infty)\).
- For \( g(x) \): The cube root function can produce all real values. Hence, \( g(x) \) can output values by shifting down by \(2\), so the range of \( g(x) \) is also all real numbers, or \((- \infty, \infty)\).
Conclusion: The ranges are not the same.
4. **X-Intercepts**:
- For \( f(x) \): The x-intercept occurs when \( f(x) = 0 \).
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3
\]
- For \( g(x) \): The x-intercept occurs when \( g(x) = 0 \).
\[
\sqrt[3]{x + 1} - 2 = 0 \implies \sqrt[3]{x + 1} = 2 \implies x + 1 = 8 \implies x = 7
\]
Conclusion: The x-intercepts are different: \( f(x) \) has an x-intercept at \( x = 3 \) and \( g(x) \) at \( x = 7 \).
Based on the analysis:
- **The true statement is**: **b. they have the same y-intercept.**