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Use the table to answer the question.

x f(x)=x3+1
g(x)=x+1

−2
−7
−1

−1
0 0
0 1 1
1 2 2
2 9 3
Compare the functions f(x)=x3+1
and g(x)=x+1
. Which of the following statements is true about the intercepts of the functions?

(1 point)
Responses

The x- and y-intercepts are different for each function.
The x - and y -intercepts are different for each function.

The graphs intersect at (1,2)
.
The graphs intersect at left parenthesis 1 comma 2 right parenthesis .

Both functions have a y-intercept of (−1,0)
and an x-intercept of (0,1)
.
Both functions have a y -intercept of left parenthesis negative 1 comma 0 right parenthesis and an x -intercept of left parenthesis 0 comma 1 right parenthesis .

Both functions have an x-intercept of (−1,0)
and a y-intercept of (0,1)
.

When comparing linear, quadratic, cubic, and exponential functions, what key feature do the all four function types share?(1 point)
Responses

The x-intercept of all four function types is (0,0)
.
The x -intercept of all four function types is left parenthesis 0 comma 0 right parenthesis .

The minimum value of all function types is (0,0)
.
The minimum value of all function types is left parenthesis 0 comma 0 right parenthesis .

The domain of all four function types is all real numbers.
The domain of all four function types is all real numbers.

The range of all four function types is all real numbers.

Compare the graphs and determine which one could be the graph of a square root function.(1 point)
Responses

A coordinate plane's axes range from negative 3 to 3, both by 1-unit increments. An S-shaped curve is plotted passing through 3 marked points.
Image with alt text: A coordinate plane's axes range from negative 3 to 3, both by 1-unit increments. An S-shaped curve is plotted passing through 3 marked points.

A coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A straight line is plotted passing through 3 marked points in the first and third quadrants.
Image with alt text: A coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A straight line is plotted passing through 3 marked points in the first and third quadrants.

A coordinate plane's x-axis ranges from negative 2 to 6 and its y-axis ranges from negative 2 to 4, both by 1-unit increments. A curve is plotted in the first quadrant. It passes through 3 marked points.
Image with alt text: A coordinate plane's x-axis ranges from negative 2 to 6 and its y-axis ranges from negative 2 to 4, both by 1-unit increments. A curve is plotted in the first quadrant. It passes through 3 marked points.

A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 1 to 4, both by 1-unit increments. A concave up parabola is drawn passing through 3 marked points in the first and second quadrants.
Image with alt text: A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 1 to 4, both by 1-unit increments. A concave up parabola is drawn passing through 3 marked points in the first and second quadrants.
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Question
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 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 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 domain, x>0
.
Both functions have the same domain, x greater than 0 .

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

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

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 .

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 .
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1 answer

Let's analyze each question one by one.

1. Compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \). Which of the following statements is true about the intercepts of the functions?

  1. Calculating the intercepts:

    • For \( f(x) \):

      • y-intercept (when \( x = 0 \)): \( f(0) = 0^3 + 1 = 1 \) → \( (0, 1) \)
      • x-intercept (when \( f(x) = 0 \)): Solve \( x^3 + 1 = 0 \) → \( x^3 = -1 \) → \( x = -1 \) → \( (-1, 0) \)

    • For \( g(x) \):

      • y-intercept (when \( x = 0 \)): \( g(0) = 0 + 1 = 1 \) → \( (0, 1) \)
      • x-intercept (when \( g(x) = 0 \)): Solve \( x + 1 = 0 \) → \( x = -1 \) → \( (-1, 0) \)

Based on this:

  • Both functions have the same y-intercept at \( (0, 1) \) and the same x-intercept at \( (-1, 0) \). Thus, the statement "Both functions have a y-intercept of (−1, 0) and an x-intercept of (0, 1)" is incorrect.

Correct statement: The x- and y-intercepts are the same for both functions.

2. When comparing linear, quadratic, cubic, and exponential functions, what key feature do the four function types share?

Among the choices, the correct one:

  • The domain of all four function types is all real numbers.

3. Compare the graphs and determine which one could be the graph of a square root function.

The characteristic of a square root function is that it has a starting point at \( (0,0) \) and only exists for non-negative x, typically increasing in the first quadrant.

From the provided descriptions:

  • The graph that fits the description of a square root function would be "A coordinate plane's x-axis ranges from negative 2 to 6 and its y-axis ranges from negative 2 to 4, both by 1-unit increments. A curve is plotted in the first quadrant. It passes through 3 marked points."

4. Compare the graphs of \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \). Which one of the following options is true?

  1. Domain:
    • \( f(x) = \sqrt{x} + 1 \) has the domain \( x \geq 0 \).
    • \( g(x) = \sqrt[3]{x} + 1 \) has the domain \( x , \in , (-\infty, \infty) \).

This means the graphs indeed have different domains.

5. Compare the graphs of \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \).

  1. \( f(x) = 2\sqrt{x} \) has a domain of \( x \geq 0 \).
  2. \( g(x) = 2\sqrt[3]{x} \) has a domain of \( x \in (-\infty, \infty) \).

Thus, the correct response is: The graphs have different domains.

In summary, the answers are:

  1. The x- and y-intercepts are the same for both functions.
  2. The domain of all four function types is all real numbers.
  3. A curve is plotted in the first quadrant.
  4. The graphs have different domains.
  5. The graphs have different domains.