A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 3 to 4, both by 1-unit increments. A dotted curve labeled g of x and a solid curve labeled f of x are plotted.

Question

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

(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
.
Both functions have the same domain, x greater than 0 .
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Answer 1

Let's analyze the two functions given:

1. \( f(x) = \sqrt{x} + 1 \)
2. \( g(x) = \sqrt[3]{x} + 1 \)

Domains

• Domain of \( f(x) \): The function \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of \( f(x) \) is \( x \geq 0 \) or in interval notation, \([0, \infty)\).

• Domain of \( g(x) \): The function \( \sqrt[3]{x} \) is defined for all real numbers. Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).

Conclusion about domain: The statement that "Both functions have the same domain, \( x > 0 \)" is false. They have different domains.

Increasing Behavior

• Increasing tendency of \( f(x) \): The derivative of \( f(x) \) is \( f'(x) = \frac{1}{2\sqrt{x}} \). Since this derivative is positive for all \( x > 0 \), \( f(x) \) is increasing on its domain \( [0, \infty) \).

• Increasing tendency of \( g(x) \): The derivative of \( g(x) \) is \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \). This derivative is also positive for all \( x \) (including negative values), so \( g(x) \) is increasing on its entire domain \( (-\infty, \infty) \).

Conclusion about increasing behavior: The statement "Both functions are increasing on their domain" is true.

Points of intersection

To check if both functions go through the points (-1, 0) and (0, 1):

• For \( f(x) \):

  • \( f(-1) \) is not defined because \( \sqrt{-1} \) is not a real number.
  • \( f(0) = \sqrt{0} + 1 = 1 \).

• For \( g(x) \):

  • \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \).
  • \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1 \).

Conclusion about points: The statement "Both functions go through the points (-1, 0) and (0, 1)" is false because \( f(x) \) does not pass through (-1, 0).

Ranges

• Range of \( f(x) \): As \( x \) approaches 0, \( f(x) \) approaches 1 and as \( x \) increases, \( f(x) \) approaches infinity. Thus, the range of \( f(x) \) is \([1, \infty)\).

• Range of \( g(x) \): As \( x \) approaches negative infinity, \( g(x) \) approaches negative infinity, and as \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity. Thus, the range of \( g(x) \) is \((- \infty, \infty)\).

Conclusion about range: The statement "Both functions have the same range, (-∞, ∞)" is false. They have different ranges.

Final Assessment

• The only true statement among the options given is "Both functions are increasing on their domain."