The graph of f(x) = |x| is stretched by a factor of 0.3 and translated down 4 units. Which statement about the domain and range of each function is correct?The range of the transformed function and the parent function are both all real numbers greater than or equal to 4.The domain of the transformed function is all real numbers and is, therefore, different from that of the parent function.The range of the transformed function is all real numbers greater than or equal to 0 and is, therefore, different from that of the parent function.The domain of the transformed function and the parent function are both all real numbers.
2. The graph of g(x) is shown.
On a coordinate plane, a piecewise function has 2 lines. The first line has a closed circle at (negative 4, 4) and goes down to a closed circle at (negative 1, 1). The second line has a closed circle at (0, 1) and goes do to an open circle at (3, negative 1).
Which statements describe the domain and range of g(x)? Select two options.
The function g(x) is defined for all real numbers x.
The maximum value of the range is 4.
The maximum value of the domain is 3.
The range of g(x) is {y| –1 < y ≤ 4}.
The domain of g(x) is {x| –4 < x ≤ 3}.
3. The graph represents the cost of catering a dinner as a function of the number of guests.On a coordinate plane, a graph titled Dinner Catering Cost shows Guests Served on the x-axis and Total cost in dollars on the y-axis. A piecewise function has 3 lines. The first line has an open circle at (0, 100) and goes up to a closed circle at (40, 600). The second line has an open circle at (40, 700) and goes up to a closed circle at (80, 1300). The third line has an open circle at (80, 1600) and goes up to (100, 2000).What is the catering cost for a dinner with 40 guests?
$400
$600
$700
$800
4.
What is the domain of the function f(x) = 3|x + 4| + 1?
all real numbers
all real numbers less than or equal to −4
all real numbers greater than or equal to 1
all real numbers greater than or equal to −4
5. Which describes the range of the parent absolute value function?
{y | y is a real number}
{y | y is an integer}
{y|y > 0}
{y|y < 0}
6. The graph of f(x) = |x| is transformed to g(x) = |x + 1| – 7. On which interval is the function decreasing?
(–∞, –7)
(–∞, –1)
(–∞, 1)
(–∞, 7)
7. The function D(t) defines a traveler’s distance from home, in miles, as a function of time, in hours.Which times and distances are represented by the function? Select three options.
D(t)= {300t+125, 0less than or equal to t < 2.5
{875, 2.5less than or equal to t less than or equal to 3.5
{75t + 612.5, 3.5<t less than or equal to 6
The starting distance, at 0 hours, is 300 miles.
At 2 hours, the traveler is 725 miles from home.
At 2.5 hours, the traveler is still moving farther from home.
At 3 hours, the distance is constant, at 875 miles.
The total distance from home after 6 hours is 1,062.5 miles.
8. SKIP
9.
The graph of f(x) = |x| is reflected across the x-axis and translated to the right 6 units. Which statement about the domain and range of each function is correct?
Both the domain and range of the transformed function are the same as those of the parent function.
Neither the domain nor the range of the transformed function are the same as those of the parent function.
The range of the transformed function is the same as the parent function, but the domains of the functions are different.
The domain of the transformed function is the same as the parent function, but the ranges of the functions are different.
10 skip.
11. What is the vertex of the graph of f(x) = |x + 3| + 7?
(3, 7)
(7, 3)
(–3, 7)
(7, –3)
12. Which is the graph of g(x)?g(x) = StartLayout Enlarged left-brace 1st row 1st column 3, 2nd column x less-than negative 2 2nd row 1st column negative StartFraction x Over 2 EndFraction + 2, 2nd column negative 2 less-than-or-equal-to x less-than 2 3rd row 1st column 2 x minus 3, 2nd column x greater-than-or-equal-to 2 EndLayout 1. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, 1) and goes up through (3, 3). 2. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, negative 5). The third line starts at (2, negative 5) and goes up to (5, 1). 3. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, negative 3) and goes up through (4, 1). 4. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, 1). The third line starts at (2, 1) and goes up through (3, 3).
13. Which is the graph of the piecewise function f(x)?f(x) = StartLayout Enlarged left-brace 1st row 1st column negative x + 1, 2nd column x less-than-or-equal-to 0 2nd row 1st column x + 1, 2nd column x greater-than 0 EndLayout 1. On a coordinate plane, a piecewise function has 2 connecting lines. The first line goes from (4, 5) down to (0, 1). The second line goes from (0, 1) up through (4, 5). 2. On a coordinate plane, a piecewise function has 2 connecting lines. The first line goes from (4, 5) down to (1, 0). The second line goes from (1, 0) up through (4, 3). 3. On a coordinate plane, a piecewise function has 2 connecting lines. The first line goes from (negative 5, negative 4) up to (0, 1). The second line goes from (0, 1) down through (4, negative 3). 4. On a coordinate plane, a piecewise function has 2 connecting lines. The first line goes from (negative 4, negative 5) up to (1, 0). The second line goes from (1, 0) down through (4, negative 3).
14. The function g(x) is defined as shown. StartLayout enlarged left-brace 1st row 1st column 3x minus 2, 2nd column negative 4 less-than-or-equal-to x less-than negative 2 2nd row 1st column negative x minus 1, 2nd column negative 2 less-than-or-equal-to x less-than 1 3rd row 1st column 6, 2nd column 1 less-than-or-equal-to x less-than-or-equal-to 3 EndLayoutWhat is the value of g(0)?
−2
−1
3
6
5 answers
Let's address each question one by one.
Transformation of f(x) = |x|:
The transformed function is \( g(x) = 0.3|x| - 4 \).
Domain: The domain of \( f(x) = |x| \) is all real numbers, and it remains the same for the transformed function. Thus, the domain is \( (-\infty, \infty) \).
Range: For \( f(x) = |x| \), the range is \( [0, \infty) \). For the transformed function, the range becomes \( [-4, \infty) \) because it is stretched vertically and translated down.
Correct Statement: The domain of the transformed function and the parent function are both all real numbers.
Graph of g(x):
Domain: The function g(x) is defined from \( x = -4 \) to \( x = 3 \), so it does not include values outside of this interval.
Range: The highest point is 4 (at the first interval), and the lowest point is -1 (not included since it ends with an open circle). Thus, ranges from -1 to 4 with -1 not included.
Correct Statements:
The maximum value of the range is 4.
The range of g(x) is {y | -1 < y ≤ 4}.
Catering Cost for 40 Guests:
At \( x = 40 \), the second segment begins from \( (40, 700) \) and ends with \( (80, 1300) \). Hence, there is an open circle at (40, 700), meaning the cost when there are exactly 40 guests is 700.
Cost: $700.
Domain of f(x) = 3|x + 4| + 1:
The function is defined for all real numbers, as it can take any value of \( x \).
Domain: All real numbers.
Range of Parent Absolute Value Function:
The absolute value function \( f(x) = |x| \) has a minimum value of 0 and extends to infinity.
Correct Range Description: {y | y ≥ 0}.
Interval where g(x) is decreasing:
The function \( g(x) = |x + 1| - 7 \) is the same as \( |x| - 6 \) when considering a horizontal translation.
The function will decrease until the vertex occurs at \( x = -1 \).
Interval of Decrease: (–∞, –1).
D(t) Function Evaluation:
Correct Statements:
At 2.5 hours, the distance is constant at 875 miles.
At 3 hours, the distance is constant, at 875 miles.
The total distance from home after 6 hours is \( 75(6) + 612.5 = 1125 + 612.5 = 1737.5 \) not shown in options, the other statements are partially true about distances at 2 hours.
Skip.
Transformation of f(x) = |x|:
The transformation results in \( g(x) = -|x - 6| \).
The domain of both functions remains all real numbers, but the range of the transformed function is different (it is upside down). The range for \( g(x) \) will be \( (-\infty, 0] \).
Correct Statement: The domain of the transformed function is the same as the parent function, but the ranges of the functions are different.
Skip.
Vertex of f(x) = |x + 3| + 7:
The vertex occurs where \( x = -3 \) and the value is \( y = 7 \).
Vertex: (–3, 7).
Graph of g(x):
To determine which graph is correct, look for the piecewise definitions:
For \( x < -2 \), g(x) is constant at 3.
From \( -2 \) to \( 2 \), it decreases to 0 (negative slope).
From \( 2 \) onwards, it is increasing.
The correct graph: should start constant, then down to \( -2 \) at \( y=0 \), and rise beyond that.
Graph of f(x):
Check the piecewise definition:
For \( x ≤ 0 \), it is decreasing.
For \( x > 0 \), it has a positive slope.
The correct graph: starts at \( (0, 1) \) down and at positive side up.
Value of g(0):
For \( g(0) \), use the second definition which is \( -x - 1 \). Plugging in 0 gives \( -0 - 1 = -1 \).
Value: -1.
Let me know if you need further clarification on any of the questions!
for 12 and 13 what graph? like 1st 2nd 3rd 4th?
Sure! Let's clarify the graphs for questions 12 and 13 based on their descriptions.
12. Which is the graph of \( g(x) \)?
The function is given as:
\[
g(x) =
\begin{cases}
3 & \text{for } x < -2 \
-\frac{x}{2} + 2 & \text{for } -2 \leq x < 2 \
2x - 3 & \text{for } x \geq 2
\end{cases}
\]
From the conditions:
For \( x < -2 \): The function is constant at 3.
From \( x = -2 \) to \( x = 2 \): The slope is negative, starting at the point (−2, 2) (since plugging in -2 into \(-\frac{x}{2} + 2\) gives 2) and ending at (2, 0).
For \( x \geq 2 \): The function begins the point (2, 1) and rises with a slope of 2.
To determine the correct graph:
The first segment should be a horizontal line at \( y = 3 \) until \( x = -2 \).
The second segment should change slope downwards until it reaches \( (2, 0) \).
The last part must start at \( (2, 1) \) and rise.
Based on these conditions, select a graph that matches this order and slope.
13. Which is the graph of the piecewise function \( f(x) \)?
The function is given as:
\[
f(x) =
\begin{cases}
-x + 1 & \text{for } x \leq 0 \
x + 1 & \text{for } x > 0
\end{cases}
\]
From the conditions:
For \( x \leq 0 \): The function will start at (0, 1) and have a negative slope, going down to the left.
For \( x > 0 \): The function will have a positive slope, starting from (0, 1) and rising to the right.
To determine the correct graph:
From the left side of the y-axis (x < 0), there should be a line falling towards the left.
At \( x = 0 \), the function should meet at (0, 1).
To the right of the y-axis (x > 0), the function should rise linearly.
Based on these conditions, select a graph that follows this description.
Summary
You need to choose the graph that meets these properties for both questions based on the given options labeled (1, 2, 3, 4). If you provide the characteristics of those graphs from the images you're looking at, I can help point you toward the correct one.
3. $600
7. At 2 hours, the traveler is 725 miles from home.
and the other two you chose
10. the check sign but on -2
13. not the check sign yes the lines go up but they are the same line and on 1
Let's evaluate your points more closely based on the clarifications you've provided:
Catering Cost for 40 Guests:
You mentioned that the catering cost for 40 guests is $600. However, based on the specified piecewise graph where the first segment connects at (0, 100) and continues to (40, 600), I had indicated $700 as being the value at 40 guests. If you have access to that graph and the details confirm that at 40 guests the cost is specifically $600, then that would be correct as per that interpretation. If there's confusion, please ensure the placement of the point on the graph aligns with this cost.
Traveler’s Distance from Home:
You pointed out that at 2 hours, the traveler is 725 miles from home. In the given piecewise function:
At \( t = 2 \): The equation is \( D(t) = 300t + 125 \), leading to:
\[
D(2) = 300(2) + 125 = 600 + 125 = 725
\]
So, this statement is indeed correct, confirming the travelers' distance.
Verify Domain for Changes:
You noted "the check sign but on -2." For the function \( f(x) = 3|x + 4| + 1 \), the previous conclusion was that the domain would be all real numbers. If the graph reflects constraints at or below x = -2 based on the transformations, then that would need to be analyzed accordingly. However, typically, the domain is wide unless explicitly restricted.
Graph of the Piecewise Function:
You mentioned that the check sign isn't the right graph but pointed out that the lines go up and are similar for function ( f(x) =
\begin{cases}
-x + 1 & \text{for } x \leq 0 \
x + 1 & \text{for } x > 0
\end{cases} ). If the graph has two lines that closely resemble, indeed— check for continuity at \( (0, 1) \) and ascertain if they are distinct pieces versus one continuous line. To accurately choose, pick a graph where it distinctly indicates different behaviors at transitions.
Overall Summary:
For cost, confirm the point on the graph aligns with the stated amount ($600 or $700).
For traveler's distance, your result of 725 miles is correct.
Clarify or resolve the domain issue. It should generally be all reals unless exceptionally limited.
For piecewise function, ensure you look for a distinct change at the defined points.
If you have visuals or specific data, please verify to ensure clarity. I'm here to help you interpret any details or graphs that you are facing challenges with!
1 answer