Question
Compare the graphs of f(x)=x+1−−−−√−2
and g(x)=^3√x+1−2
. Identify which of the following features is true about the graphs.
(1 point)
Responses
The graphs have the same x-intercept.
The graphs have the same x -intercept.
The graphs have the same domains.
The graphs have the same domains.
The graphs have the same ranges.
The graphs have the same ranges.
The graphs have the same y-intercept.
The graphs have the same y -intercept.
and g(x)=^3√x+1−2
. Identify which of the following features is true about the graphs.
(1 point)
Responses
The graphs have the same x-intercept.
The graphs have the same x -intercept.
The graphs have the same domains.
The graphs have the same domains.
The graphs have the same ranges.
The graphs have the same ranges.
The graphs have the same y-intercept.
The graphs have the same y -intercept.
Answers
GPT-4o mini
Let's analyze the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = \sqrt[3]{x + 1} - 2 \) to determine the features mentioned.
1. **Finding the x-intercepts**:
- For \( f(x) \), set \( f(x) = 0 \):
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3.
\]
- For \( g(x) \), set \( g(x) = 0 \):
\[
\sqrt[3]{x + 1} - 2 = 0 \implies \sqrt[3]{x + 1} = 2 \implies x + 1 = 8 \implies x = 7.
\]
- The x-intercept of \( f(x) \) is at \( x = 3 \), while the x-intercept of \( g(x) \) is at \( x = 7 \). Therefore, they do **not have the same x-intercept**.
2. **Finding the domains**:
- The domain of \( f(x) \) requires \( x + 1 \geq 0 \) which gives \( x \geq -1 \). Hence, the domain is \( [-1, \infty) \).
- The domain of \( g(x) \) is all real numbers since cube roots are defined for all \( x \). Therefore, the domain is \( (-\infty, \infty) \).
- The domains are **not the same**.
3. **Finding the ranges**:
- For \( f(x) \), as \( x \) approaches \(-1\), \( \sqrt{-1 + 1} - 2 = 0 - 2 = -2\). As \( x \) approaches infinity, \( f(x) \to \infty \). Hence, the range is \([-2, \infty)\).
- For \( g(x) \), as \( x \) approaches \(-1\), \( \sqrt[3]{-1 + 1} - 2 = 0 - 2 = -2\). As \( x \) approaches infinity, \( g(x) \to \infty \). Therefore, the range is also \([-2, \infty)\).
- The ranges are **the same**.
4. **Finding the y-intercepts**:
- The y-intercept for \( f(x) \) occurs when \( x = 0 \):
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1.
\]
- The y-intercept for \( g(x) \) occurs when \( x = 0 \):
\[
g(0) = \sqrt[3]{0 + 1} - 2 = 1 - 2 = -1.
\]
- The y-intercepts are both \( -1 \), so they have **the same y-intercept**.
Summarizing the results:
- The graphs do **not have the same x-intercept.**
- The graphs do **not have the same domains.**
- The graphs **have the same ranges.**
- The graphs **have the same y-intercept.**
Therefore, the true statements are:
- **The graphs have the same ranges.**
- **The graphs have the same y-intercept.**
1. **Finding the x-intercepts**:
- For \( f(x) \), set \( f(x) = 0 \):
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3.
\]
- For \( g(x) \), set \( g(x) = 0 \):
\[
\sqrt[3]{x + 1} - 2 = 0 \implies \sqrt[3]{x + 1} = 2 \implies x + 1 = 8 \implies x = 7.
\]
- The x-intercept of \( f(x) \) is at \( x = 3 \), while the x-intercept of \( g(x) \) is at \( x = 7 \). Therefore, they do **not have the same x-intercept**.
2. **Finding the domains**:
- The domain of \( f(x) \) requires \( x + 1 \geq 0 \) which gives \( x \geq -1 \). Hence, the domain is \( [-1, \infty) \).
- The domain of \( g(x) \) is all real numbers since cube roots are defined for all \( x \). Therefore, the domain is \( (-\infty, \infty) \).
- The domains are **not the same**.
3. **Finding the ranges**:
- For \( f(x) \), as \( x \) approaches \(-1\), \( \sqrt{-1 + 1} - 2 = 0 - 2 = -2\). As \( x \) approaches infinity, \( f(x) \to \infty \). Hence, the range is \([-2, \infty)\).
- For \( g(x) \), as \( x \) approaches \(-1\), \( \sqrt[3]{-1 + 1} - 2 = 0 - 2 = -2\). As \( x \) approaches infinity, \( g(x) \to \infty \). Therefore, the range is also \([-2, \infty)\).
- The ranges are **the same**.
4. **Finding the y-intercepts**:
- The y-intercept for \( f(x) \) occurs when \( x = 0 \):
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1.
\]
- The y-intercept for \( g(x) \) occurs when \( x = 0 \):
\[
g(0) = \sqrt[3]{0 + 1} - 2 = 1 - 2 = -1.
\]
- The y-intercepts are both \( -1 \), so they have **the same y-intercept**.
Summarizing the results:
- The graphs do **not have the same x-intercept.**
- The graphs do **not have the same domains.**
- The graphs **have the same ranges.**
- The graphs **have the same y-intercept.**
Therefore, the true statements are:
- **The graphs have the same ranges.**
- **The graphs have the same y-intercept.**