Question

Question

Use the image to answer the 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

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−−√
.
When x greater than 1 , the function g left parenthesis x right parenthesis equals 2 root index 3 Start Root x End Root is greater than f left parenthesis x right parenthesis equals 2 Start Root x End Root .

The graphs have different domains.
The graphs have different domains.
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Answer 1

To compare the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt{x} - 3 \), let’s analyze them step by step regarding their behavior.

1. Domain:

   • Both functions involve a square root, which means their domains are the same: \( x \geq 0 \). Therefore, they do not have different domains.

2. Intercepts:

   • \( f(x) = 2\sqrt{x} \) goes through the points:
     • At \( x = -1 \): Not defined since we cannot take the square root of a negative number.
     • At \( x = 0 \): \( f(0) = 2\sqrt{0} = 0 \) → Point (0, 0).
     • At \( x = 1 \): \( f(1) = 2\sqrt{1} = 2 \) → Point (1, 2).
   • \( g(x) = 2\sqrt{x} - 3 \) goes through the points:
     • At \( x = -1 \): Not defined.
     • At \( x = 0 \): \( g(0) = 2\sqrt{0} - 3 = -3 \) → Point (0, -3).
     • At \( x = 1 \): \( g(1) = 2\sqrt{1} - 3 = -1 \) → Point (1, -1).
   • The graphs do not go through the same points at \( x = -1, 0, 1 \).

3. Increasing/Decreasing Behavior:

   • Both functions are increasing on their domains since the square root function increases as \( x \) increases. Therefore, both \( f \) and \( g \) are increasing, not decreasing.

4. Comparing values when \( x > 1 \):

   • For \( x > 1 \):
     • \( f(x) = 2\sqrt{x} \) will yield positive values starting from \( (1, 2) \) upwards.
     • \( g(x) = 2\sqrt{x} - 3 \) will yield values that remain below the x-axis after passing \( (1, -1) \) because the subtraction of 3 shifts the graph downwards.
   • Therefore, for \( x > 1\), \( g(x) \) will be less than \( f(x) \).

Based on the analysis above, the correct response to the comparison is:

When \( x > 1 \), the function \( g(x) = 2\sqrt{x} - 3 \) is less than \( f(x) = 2\sqrt{x} \). The first two statements about passing through \((-1, -1)\), \((0, 0)\), and \((1, 1)\) or both being decreasing are not accurate.

If forced to choose from the options presented, the most appropriate true statement is:

They are both decreasing on their domains (This is false; hence, it does not qualify).

None of the provided statements correctly reflect the relationship between the graphs and thus indicate a misunderstanding. The correct acknowledgment would be:

The third statement goes ill-defined as both functions truly are increasing on their common domain.

Therefore, if assessing logically what may apply to the presented relationships would yield statements reflecting those increasing nature direct statements about positioning or relationships of values where they overlap or diverge. The comparative graph movements must thus align with these fundamentals as explored through the outright examination of the defined mathematical identities.