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Question

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.

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 have the same domain, x>0
.
Both functions have the same domain, x greater than 0 .

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 are increasing on their domain.

Answer 1

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

   Conclusion: Both functions do not have the same domain.

2. Points:

   • For \( f(x) \):
     • At \( x = -1 \): \( f(-1) \) is undefined because the square root of a negative number is not a real number.
     • At \( x = 0 \): \( f(0) = \sqrt{0} + 1 = 1 \).
   • For \( g(x) \):
     • At \( x = -1 \): \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \).
     • At \( x = 0 \): \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1 \).

   Conclusion: The functions do not go through the points \( (-1, 0) \) and \( (0, 1) \); \( f(x) \) is undefined at \( x = -1 \).

3. Range:

   • \( f(x) = \sqrt{x} + 1 \): The minimum value occurs at \( x = 0 \) is \( f(0) = 1 \). As \( x \) increases, \( f(x) \) increases without bound. Thus, the range is \( [1, \infty) \).
   • \( g(x) = \sqrt[3]{x} + 1 \): The cube root function can take on any real value, and adding 1 shifts the entire graph up by 1. Therefore, the range is all real numbers: \( (-\infty, \infty) \).

   Conclusion: Both functions do not have the same range.

4. Increasing Nature:

   • Both \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \) are increasing functions in their respective domains.
   • \( f(x) \) is increasing for \( x \geq 0 \), and \( g(x) \) is increasing for all \( x \).

   Conclusion: Both functions are increasing on their domains.

After analyzing the provided options, the true statement is:

Both functions are increasing on their domain.