Use the image to answer the question.

Question

A coordinate plane's axes range from negative 4 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 given graphs of f(x)=2x−−√
and g(x)=2x−−√3
and determine which of the following is true.

(1 point)
Responses

The graphs have different domains.
The graphs have different domains.

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

They are both decreasing on their domains.
They are both decreasing on their domains.

When x>1
, the function g(x)=2x−−√3
is greater than f(x)=2x−−√
.

Answer 1

To compare the functions \( f(x) = \sqrt{2x} \) and \( g(x) = \sqrt{2x} + 3 \), let's analyze each statement one by one.

1. The graphs have different domains.

   • Both functions \( f(x) = \sqrt{2x} \) and \( g(x) = \sqrt{2x} + 3 \) only exist for \( x \geq 0 \) since the square root function is only defined for non-negative numbers. Thus, their domains are the same: \( [0, \infty) \). This statement is false.

2. Both graphs go through (−1,−1), (0,0), and (1,1).

   • For \( f(x) = \sqrt{2x} \):
     • At \( x=0 \): \( f(0) = \sqrt{2(0)} = 0 \) (so, it goes through (0, 0)).
     • At \( x=1 \): \( f(1) = \sqrt{2(1)} = \sqrt{2} \), approximately 1.414 (so, it does not go through (1, 1)).
     • At \( x=-1 \): \( f(-1) \) is not defined (so, it does not go through (-1, -1)).
   • For \( g(x) = \sqrt{2x} + 3 \):
     • At \( x=0 \): \( g(0) = \sqrt{2(0)} + 3 = 3 \).
     • At \( x=1 \): \( g(1) = \sqrt{2(1)} + 3 = \sqrt{2} + 3 \), approximately 4.414 (not (1, 1)).
     • At \( x=-1 \): \( g(-1) \) is also not defined.
   • Thus, this statement is false.

3. They are both decreasing on their domains.

   • For \( f(x) = \sqrt{2x} \) and \( g(x) = \sqrt{2x} + 3 \), both functions are increasing as \( x \) increases. Therefore, this statement is false.

4. When \( x>1 \), the function \( g(x) = \sqrt{2x} + 3 \) is greater than \( f(x) = \sqrt{2x} \).

   • For \( x > 1 \):
     • Since \( g(x) = \sqrt{2x} + 3 \) adds a constant (3) to \( f(x) \), it is clear that \( g(x) \) is always greater than \( f(x) \) for \( x > 1 \). This statement is true.

Based on the evaluations, the correct answer is:

When x>1, the function g(x)=√(2x)+3 is greater than f(x)=√(2x).