Jillian walked 0.5 miles before she started jogging at an average pace of 5 miles per hour. The equation d = 0.5 + 5t can be used to relate the total distance, d, in miles to the time, t, that Jillian spent jogging. What are the independent and dependent variables?

Question

The independent variable is distance and the dependent variable is time.
The independent variable is time and the dependent variable is distance.
The independent variable is distance and the dependent variable is speed.
The independent variable is speed and the dependent variable is distance.

Answer 1

In the equation \( d = 0.5 + 5t \), the variables relate as follows:

• \( t \) (time spent jogging) is the independent variable because you can choose how long Jillian jogs.
• \( d \) (total distance) is the dependent variable because the total distance depends on the time she spends jogging.

Therefore, the correct answer is:

The independent variable is time and the dependent variable is distance.

Answer 2

Which graph represents a function?

On a coordinate plane, a straight vertical line crosses the y-axis 0 times and the x-axis 1 time.

On a coordinate plane, a curved line crosses the y-axis 2 times and the x-axis 1 time.

On a coordinate plane, 2 curved lines are shown. Both lines touch the y-axis 1 time and the x-axis 1 time.

On a coordinate plane, a straight horizontal line crosses the y-axis 1 time and the x-axis 0 times.

Answer 3

To determine which graph represents a function, we can use the vertical line test. A graph is a function if no vertical line can intersect the graph at more than one point.

1. Vertical Line: A straight vertical line that crosses the x-axis at one point does not represent a function because a vertical line is not a function.
2. Curved Line: A curved line that crosses the y-axis twice means it can be intersected by a vertical line at more than one point, so it is not a function.
3. Two Curved Lines: If there are two curved lines that touch the y-axis and the x-axis, if they are separate and do not overlap, each could be a function, but we need further details about their shapes to conclude.
4. Horizontal Line: A horizontal line that crosses the y-axis at one point represents a function. A horizontal line does not violate the vertical line test because a vertical line will intersect it at most one point.

Based on this analysis, the graph representing a horizontal line that crosses the y-axis 1 time and the x-axis 0 times is the best choice.

Thus, the correct answer is that the straight horizontal line represents a function.

Answer 4

The set of ordered pairs (–1, 8), (0, 3), (1, –2), and (2, –7) represent a function. What is the range of the function?

{x: x = –1, 0, 1, 2}
{y: y = –7, –2, 3, 8}
{x: x = –7, –2, –1, 0, 1, 2, 3, 8}
{y: y = –7, –2, –1, 0, 1, 2, 3, 8}

Answer 5

In this context, the range of a function consists of all the possible output values (y-values) from the ordered pairs.

The given ordered pairs are:

• (–1, 8)
• (0, 3)
• (1, –2)
• (2, –7)

The y-values are:

• 8 (from the first pair)
• 3 (from the second pair)
• –2 (from the third pair)
• –7 (from the fourth pair)

Now, listing the unique y-values, we get the range:

• 8
• 3
• –2
• –7

Thus, the range of the function is \( {y: y = –7, –2, 3, 8} \).

So, the correct answer is: {y: y = –7, –2, 3, 8}.

Answer 6

On a coordinate plane, a curved line crosses the x-axis at (negative 1.5, 0), the y-axis at (0, negative 2), and the x-axis at (1.5, 0).
Determine the domain and range of the given function.

The domain is
.

The range is
.

Answer 7

To determine the domain and range of the function represented by the described curved line, we'll analyze the provided points where it intersects the axes.

Points Provided:

1. Intersects the x-axis at \((-1.5, 0)\) and \((1.5, 0)\).
2. Intersects the y-axis at \((0, -2)\).

Domain:

The domain of a function is the set of all possible x-values for which the function is defined. From the x-axis intersections:

• The curve crosses the x-axis at \(-1.5\) and \(1.5\).
• It does not seem to extend beyond these points based on the given information.

Thus, the domain can be expressed as:

• From \(-1.5\) to \(1.5\).

Domain: \([-1.5, 1.5]\)

Range:

The range of a function is the set of all possible y-values that the function can take.

1. The curve intersects the y-axis at \(-2\).
2. Since it intersects the x-axis at \(0\) (both points on the x-axis are \(0\)), and based on the nature of the curve (typically a continuous path), it can be inferred that:
   • The lowest y-value appears to be \(-2\) (the y-intercept).
   • The highest y-value occurs at \(0\) (the x-axis intersections).

Thus, the range is:

• From \(-2\) to \(0\).

Range: \([-2, 0]\)

Final Answer:

• The domain is \([-1.5, 1.5]\).
• The range is \([-2, 0]\).

Answer 8

the option for the first one is {x:x=−2,−1,0,1,2} and {y:y=−2,−1,0,1,2} and all real numbers

the second one the options: {y:y=−2,−1,0,1,2} all real numbers greater than or equal to -2 and all real numbers

Answer 9

Based on the options you provided for the domain and range of the function you described previously, we need to identify the correct sets given the characteristics of the function.

Recap of Points:

• The function crosses the x-axis at \((-1.5, 0)\) and \((1.5, 0)\).
• The function crosses the y-axis at \((0, -2)\).

Assessing the Domain:

1. Domain: The curve spans the x-values from \(-1.5\) to \(1.5\). However, if the options were limited to specific integers, then the closest integers surrounding the domain could be chosen.

   • The given option \({x : x = -2, -1, 0, 1, 2}\) includes integer values within this interval and might be the choice presented given the context of bounding it with integers, although it doesn’t match exactly with the more precise interval of \([-1.5, 1.5]\).

   If you have to choose from the options listed, select:

   • {x : x = −2, −1, 0, 1, 2}.

Assessing the Range:

2. Range: The lowest point on the curve occurs at \(y = -2\), and it does intersect the x-axis at \(y = 0\).
   • Looking at the options, \({y : y = −2, −1, 0, 1, 2}\) includes values from the lowest point to the x-axis.
   • The other option given, "all real numbers greater than or equal to -2", is also relevant because the range starts at \(-2\) and extends upwards.

If you want a specific set or a fewer number of values, select:

• {y : y = −2, −1, 0, 1, 2}.

If you are looking for a concise answer based on the provided options, here’s the summary:

• Domain: \({x : x = −2, −1, 0, 1, 2}\)
• Range: \({y : y = −2, −1, 0, 1, 2}\)

If you are considering a more accurate representation that might not exactly match, you can refer to "all real numbers greater than or equal to -2" for the range as a broader classification.

Answer 10

Determine the domain and range of the given function.

The domain is
✔ all real numbers
.

The range is
✔ all real numbers greater than or equal to –2
.

The mapping shows a relationship between input and output values.

A mapping diagram shows a relation, using arrows, between input and output for the following ordered pairs: (negative 5, 0), (2, negative 3), (negative 1, negative 3), (4, negative 2), (4, 2), (6, negative 1).
Which ordered pair could be removed to make this relation a function?

(–5, 0)
(–1, –3)
(4, –2)
(6, –1)

Answer 11

To determine which ordered pair could be removed to make the relation a function, we need to identify any repeated inputs (or x-values) because a relation is a function if each input is associated with exactly one output.

Given Ordered Pairs:

• (–5, 0)
• (2, –3)
• (–1, –3)
• (4, –2)
• (4, 2) (This has the same x-value as another pair)
• (6, –1)

Analysis:

• The ordered pair (4, –2) and (4, 2) both have the same input (x-value of 4) but different outputs (–2 and 2).
• This means that the relation is not a function due to this repetition.

Options to Remove:

• Removing (4, –2) would still leave (4, 2), which would allow the relation to be a function.
• Removing (4, 2) would still leave (4, –2), which would also still allow the relation to be a function.

The remaining pairs:

• (–5, 0)
• (2, –3)
• (–1, –3)
• (6, –1)

Conclusion:

To make the relation a function, you could remove either one of the pairs with the input of 4. The question asks for an option from the listed pairs:

• Remove either (4, –2) or (4, 2) to make the relation a function, but since these options aren't provided, you would need to choose based on the other options given, which do not lead to any ambiguity.

Since the pairs provided to choose from are:

• (–5, 0)
• (–1, –3)
• (4, –2)
• (6, –1)

Out of these options, since we need to maintain the unique mapping, we can pick:

• (4, –2) or (4, 2) specifically mentioned in your original question for removal, but since (4, –2) is on the list, one option to remove is:

(4, –2) or (4, 2), whichever fits your requirements better.