Use the image to answer the question.

Question

A curve labeled y equals 4 root index 3 start root x minus 2 end root is plotted on a coordinate plane. The curve begins in quadrant 3 sloping upward from left to right, briefly passes through quadrant 4, and rises through quadrant 1.

Which description matches the graph of the function?

(1 point)
Responses

The function increases as x
increases.
The function increases as x increases.

The function first decreases, then increases as x
increases.
The function first decreases, then increases as x increases.

The function first increases, then decreases as x
increases.
The function first increases, then decreases as x increases.

The function decreases as x
increases.

Use the table to answer the question.

Option #1 Option #2 Option #3 Option #4
The functions have the same domain. The functions have different domains. The functions have different domains. The functions have different domains.
The functions have the same range. The functions have different ranges. The functions have the same range. The functions have different ranges.
Both functions are decreasing. Both functions are decreasing. Both functions are increasing. Both functions are increasing.
Compare the graphs of f(x)=−2x−−√
and g(x)=−2x−−√3
. Which of the options describing the graphs is true?(1 point)
Responses

Option #1
Option #1

Option #3
Option #3

Option #2
Option #2

Option #4

Use the image to answer the question.

A dotted line and a solid curve are plotted on a coordinate plane. The line and the curve intersect on the y-axis and again in quadrant 1.

Compare the functions graphed. What is true about the functions?

(1 point)
Responses

One is a curve and the other is a line, but the y
-values of both functions increases as x
increases.
One is a curve and the other is a line, but the y-values of both functions increases as x increases.

The curved function is defined only for values of x
on the interval [−5,2]
. The straight function is defined only for values of x
on the interval [−2,3]
.
The curved function is defined only for values of x on the interval left bracket negative 5 comma 2 right bracket. The straight function is defined only for values of x on the interval left bracket negative 2 comma 3 right bracket.

The curved function has two y
-intercepts, but the straight function has only 1.
The curved function has two y-intercepts, but the straight function has only 1.

For large values of x
, both functions appear as straight lines.

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

Both functions are increasing.
Both functions are increasing.

Both functions have the same domain, [0,∞)
.
Both functions have the same domain, left bracket 0 comma infinity right parenthesis .

Both functions have the same range, (−∞,∞)
.
Both functions have the same range, left parenthesis negative infinity comma infinity right parenthesis .

Both functions go through the point (−1,−1)
.

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 2 to 2, 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)=x−−√
and g(x)=x−−√3
, and determine which of the following statements is true.

(1 point)
Responses

The functions intersect at the points (−1,−1)
, (0,0)
, and (1,1)
.
The functions intersect at the points left parenthesis negative 1 comma negative 1 right parenthesis , left parenthesis 0 comma 0 right parenthesis , and left parenthesis 1 comma 1 right parenthesis .

The functions have the same range.
The functions have the same range.

Both functions are always increasing.
Both functions are always increasing.

The functions have the same domains.

Answer 1

Let's analyze the questions and their options one by one.

1. Graph of the Function y = 4√(3√(x - 2)):

   • The description states that the curve increases from quadrant 3 through quadrant 4 and rises through quadrant 1, indicating that as \( x \) increases, \( y \) also increases.
   • Therefore, the correct response is: The function increases as x increases.

2. Comparison of Functions f(x) = -2√x and g(x) = -2√(3):

   • Both options differ in their range and domain as the expression inside the square root has different forms.
   • Given that both functions are decreasing, and since they have the same general structure, we should look at the options for the characteristics mentioned.
   • The response that seems most accurate based on the descriptions is Option #1, which states: The functions have the same domain.

3. Comparison of a Curved Function and a Straight Line:

   • The dotted curve and solid line intersect on the y-axis and again in quadrant 1 and are distinct types of graphs.
   • Since one function is linear and the other is a curve, the option that correctly explains their y-value behavior is: One is a curve and the other is a line, but the y-values of both functions increase as x increases.

4. Comparison of Functions f(x) = √x and g(x) = √(3):

   • For these functions, the typical characteristics state that both functions go through the point (0,0) and retain the domain as [0,∞).
   • The correct statement among the options is: Both functions are increasing.

5. Final Comparison of f(x) = √x and g(x) = √(3):

   • Looking at the provided responses and the previous justifications, we can conclude that both functions are indeed always increasing and go through the same origin point due to the properties of square root functions.
   • Hence, the best response to this query is: Both functions are always increasing.

To summarize the answers to the questions:

1. The function increases as x increases.
2. Option #1
3. One is a curve and the other is a line, but the y-values of both functions increases as x increases.
4. Both functions are increasing.
5. Both functions are always increasing.