compare the graphs of f (x) = √x + 1 and g (x) = ∛x + 1. which one of the following options is true?

Answer 1

To compare the graphs of the functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \), let's analyze their properties:

1. Domain:

   • \( f(x) = \sqrt{x} + 1 \): The square root function is defined for \( x \geq 0 \). Thus, the domain of \( f \) is \( [0, \infty) \).
   • \( g(x) = \sqrt[3]{x} + 1 \): The cube root function is defined for all real numbers. So, the domain of \( g \) is \( (-\infty, \infty) \).

2. Range:

   • \( f(x) \): Since the minimum value is at \( x = 0 \), where \( f(0) = 1 \), the range is \( [1, \infty) \).
   • \( g(x) \): The function can take any real number value, as \( g(x) \) approaches \( -\infty \) as \( x \) approaches \( -\infty \) and approaches \( +\infty \) as \( x \) approaches \( +\infty \). The range is \( (-\infty, \infty) \).

3. Behavior as \( x \) approaches specific values:

   • As \( x \to 0 \):
     • \( f(0) = \sqrt{0} + 1 = 1 \)
     • \( g(0) = \sqrt[3]{0} + 1 = 1 \)
   • As \( x \to \infty \):
     • \( f(x) \to \infty \)
     • \( g(x) \to \infty \)
   • As \( x \to -\infty \):
     • \( f(x) \) is not defined.
     • \( g(x) \to -\infty \)

4. Shape of the graphs:

   • \( f(x) \): The graph of \( f(x) = \sqrt{x} + 1 \) starts at the point (0, 1) and increases slowly as \( x \) increases.
   • \( g(x) \): The graph of \( g(x) = \sqrt[3]{x} + 1 \) passes through the origin at (0, 1), has the typical S-shape of a cubic root function, and extends indefinitely in both positive and negative directions.

Conclusion:

Based on the above analysis:

• The function \( f(x) \) is defined only for \( x \geq 0 \), while \( g(x) \) is defined for all \( x \).
• The range of \( f \) is limited to values \( y \geq 1 \), while \( g \) spans all real numbers.
• Both graphs meet at the point (0, 1), and as \( x \) increases, \( f(x) \) increases less steeply than \( g(x) \).

True statements:

Based on these comparisons, the following statements can be true:

1. The function \( f(x) \) is only defined for \( x \geq 0 \), whereas \( g(x) \) is defined for all \( x \).
2. The graph of \( g(x) \) will extend infinitely in both positive and negative directions.

If you provide the specific options, I can help determine which option is true based on this analysis!