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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.

1 answer

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.