Compare the graphs of f(x)=x−−√+1 and g(x)=x−−√3+1 . Which one of the following options is true?

Question

(1 point)
Responses

Both functions are increasing on their domain.
Both functions are increasing on their domain.

Both functions go through the points (−1,0) and (0,1).
Both functions go through the points left parenthesis negative 1 comma 0 right parenthesis and left parenthesis 0 comma 1 right parenthesis .

Both functions have the same range, (−∞,∞).
Both functions have the same range, left parenthesis negative infinity comma infinity right parenthesis .

Both functions have the same domain, x>0.

Answer 1

To compare the functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \), we need to analyze their domains, ranges, and behavior.

1. Domain:

   • For \( f(x) = \sqrt{x} + 1 \), the domain is \( x \geq 0 \) since the square root function is only defined for non-negative values.
   • For \( g(x) = \sqrt[3]{x} + 1 \), the domain is all real numbers \( (-\infty, \infty) \) since the cube root function is defined for all real values.
   • Conclusion: They do not have the same domain.

2. Range:

   • For \( f(x) = \sqrt{x} + 1 \), the range begins at \( 1 \) (when \( x=0 \)) and extends to infinity, so the range is \( [1, \infty) \).
   • For \( g(x) = \sqrt[3]{x} + 1 \), as \( x \) goes from \( -\infty \) to \( \infty \), \( g(x) \) also covers all real numbers. Thus, the range is \( (-\infty, \infty) \).
   • Conclusion: They do not have the same range.

3. Increasing or Decreasing:

   • For both functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \), we can analyze their derivatives to determine their monotonicity:
     • \( f'(x) = \frac{1}{2\sqrt{x}} \), which is positive for \( x > 0 \), indicating that \( f(x) \) is increasing on its domain \( [0, \infty) \).
     • \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \), which is also positive for all real \( x \) (including negative values), indicating that \( g(x) \) is increasing on its entire domain \( (-\infty, \infty) \).
   • Conclusion: Both functions are increasing on their respective domains.

4. Points of Interest:

   • Checking the points:
     • \( f(-1) = \sqrt{-1} + 1 \) is undefined, so \( f(x) \) does not pass through \((-1, 0)\).
     • \( f(0) = \sqrt{0} + 1 = 1 \).
     • \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0\).
     • \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1\).
   • Conclusion: Both functions do not pass through the point \((-1, 0)\).

In summary:

• Both functions are increasing on their domains.
• They do not pass through \((-1, 0)\) and \( (0, 1) \).
• They do not have the same range.
• They do not have the same domain.

Therefore, the true statement among the options provided is:

Both functions are increasing on their domain.