Question

Compare the graphs of f(x) = sqrt(x + 1) - 2 and g(x) = root(x + 1, 3) - 2 Identify which of the following features is true about the graphs (1 point)
The graphs have the same domains
The graphs have the same y-intercept
The graphs have the same ranges
The graphs have the same x-intercept.

Answer 1

To compare the graphs of the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = \sqrt[3]{x + 1} - 2 \), we will analyze their domains, y-intercepts, and ranges.

1. **Domains**:
- **For \( f(x) \)**:
- The expression \( \sqrt{x + 1} \) is defined when \( x + 1 \geq 0 \), so \( x \geq -1 \).
- The domain of \( f(x) \) is \( [-1, \infty) \).
- **For \( g(x) \)**:
- The cube root function \( \sqrt[3]{x + 1} \) is defined for all real numbers.
- The domain of \( g(x) \) is \( (-\infty, \infty) \).

So, the domains are not the same.

2. **Y-Intercepts**:
- **For \( f(x) \)**:
- Evaluating at \( x = 0 \):
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1
\]
- **For \( g(x) \)**:
- Evaluating at \( x = 0 \):
\[
g(0) = \sqrt[3]{0 + 1} - 2 = 1 - 2 = -1
\]

Both functions have the same y-intercept at \( -1 \).

3. **Ranges**:
- **For \( f(x) \)**:
- The minimum value occurs at \( x = -1 \):
\[
f(-1) = \sqrt{-1 + 1} - 2 = 0 - 2 = -2
\]
- As \( x \to \infty \), \( f(x) \to \infty \).
- Thus, the range of \( f(x) \) is \( [-2, \infty) \).
- **For \( g(x) \)**:
- As \( x \) approaches \( -1 \), \( g(x) \) approaches \( -2 \). As \( x \to \infty \), \( g(x) \to \infty \).
- The range of \( g(x) \) is also \( (-2, \infty) \).

The ranges are not the same because the range of \( f(x) \) includes the endpoint \( -2 \), while the range of \( g(x) \) does not.

4. **X-Intercepts**:
- **For \( f(x) \)**:
- Setting \( f(x) = 0 \):
\[
\sqrt{x + 1} - 2 = 0 \implies \sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3
\]
- **For \( g(x) \)**:
- Setting \( g(x) = 0 \):
\[
\sqrt[3]{x + 1} - 2 = 0 \implies \sqrt[3]{x + 1} = 2 \implies x + 1 = 8 \implies x = 7
\]

Thus, the x-intercepts are different: \( f(x) \) has an x-intercept at \( x = 3 \), while \( g(x) \) has an x-intercept at \( x = 7 \).

### Summary:
- The functions do **not** have the same domains.
- The functions **do** have the same y-intercept.
- The functions do **not** have the same ranges.
- The functions do **not** have the same x-intercepts.

The only true feature is **the graphs have the same y-intercept**.