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Compare f(x)= square root x+1 and g(x) = ^3 square root x +1

Both are increasing in their domain
Both have the same range (- infinity, infinity)
Both go through points (-1,0) and (0,1)
Both have the same domain, x> 0

1 answer

Let’s analyze the two functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \) based on the claims you've made.

  1. Domain:

    • \( f(x) = \sqrt{x} + 1 \) has a domain of \( x \geq 0 \) because the square root function is defined only for non-negative real numbers.
    • \( g(x) = \sqrt[3]{x} + 1 \) has a domain of all real numbers \( (-\infty, \infty) \) because the cube root function is defined for all real numbers.
    • Therefore, the statement that both functions have the same domain \( x > 0 \) is incorrect, as \( g(x) \) is defined for all \( x \).

  2. Range:

    • \( f(x) = \sqrt{x} + 1 \) has a range of \( [1, \infty) \) because as \( x \) increases from 0 to infinity, \( f(x) \) starts at 1 (when \( x = 0 \)) and increases without bound.
    • \( g(x) = \sqrt[3]{x} + 1 \) has a range of \( (-\infty, \infty) \) because as \( x \) approaches negative infinity, \( g(x) \) also approaches negative infinity; and as \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity.
    • Thus, both functions do not have the same range, contradicting the statement.

  3. Increasing Behavior:

    • Both \( f(x) \) and \( g(x) \) are increasing functions in their respective domains.
      • \( f(x) \) is increasing for \( x \geq 0 \).
      • \( g(x) \) is increasing for all \( x \).
    • This claim is correct.

  4. Points (-1, 0) and (0, 1):

    • Neither function passes through the point (-1, 0):
      • \( f(-1) = \sqrt{-1} + 1 \) is undefined.
      • \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \).
    • The point (0, 1) is valid for both functions:
      • \( f(0) = \sqrt{0} + 1 = 1 \).
      • \( g(0) = \sqrt[3]{0} + 1 = 1 \).
    • Only the point \( (0, 1) \) is correct for both functions, while the point (-1, 0) is not valid for \( f(x) \).

Summary:

  • The functions do not have the same domain.
  • They have different ranges.
  • Both functions are indeed increasing in their respective domains.
  • They share the point (0, 1) and do not intersect at (-1, 0).

Corrected Statements:

  • The domains are \( x \geq 0 \) for \( f(x) \) and \( x \in (-\infty, \infty) \) for \( g(x) \).
  • The ranges are \( [1, \infty) \) for \( f(x) \) and \( (-\infty, \infty) \) for \( g(x) \).
  • Both are increasing in their respective domains.
  • Both pass through the point (0, 1), but only one function \( g(x) \) passes through (−1, 0).