Question
Use the graph to find the following.
-6
6
12
6
12
18
24
-6
x
y
A coordinate system has a horizontal x-axis labeled from negative 6 to 12 in increments of 1 and a vertical y-axis labeled from negative 6 to 26 in increments of 1. A horizontal ray starts at point (0, 24) and extends to the left. A curve falls from left to right starting at the point (0, 24) and passing through the point (3, 3) to the point (5, negative 1). The curve then rises from left to right passing through the point (7, 3).
(a) the domain of f
(b) the range of f
(c) the x-intercepts
(d) the y-intercept
(e) intervals on which f is increasing
(f) intervals on which f is decreasing
(g) intervals on which f is constant
(h) the number at which f has a relative minimum
(i) the relative minimum of f
(j) f(negative 5)
(k) The values of x for which f(x)equals3
(l) Is f even, odd or neither?
Question content area bottom
Part 1
(a) What is the domain of f?
enter your response here (Type your answer in interval notation.)
-6
6
12
6
12
18
24
-6
x
y
A coordinate system has a horizontal x-axis labeled from negative 6 to 12 in increments of 1 and a vertical y-axis labeled from negative 6 to 26 in increments of 1. A horizontal ray starts at point (0, 24) and extends to the left. A curve falls from left to right starting at the point (0, 24) and passing through the point (3, 3) to the point (5, negative 1). The curve then rises from left to right passing through the point (7, 3).
(a) the domain of f
(b) the range of f
(c) the x-intercepts
(d) the y-intercept
(e) intervals on which f is increasing
(f) intervals on which f is decreasing
(g) intervals on which f is constant
(h) the number at which f has a relative minimum
(i) the relative minimum of f
(j) f(negative 5)
(k) The values of x for which f(x)equals3
(l) Is f even, odd or neither?
Question content area bottom
Part 1
(a) What is the domain of f?
enter your response here (Type your answer in interval notation.)
Answers
GPT-4o mini
To determine the properties of the function \( f \) described by the graph, here's how we would analyze each part:
### (a) The domain of \( f \)
Based on the description and the coordinate system provided, the graph is defined from \( x = -6 \) to \( x = 12 \). Therefore, the domain in interval notation is:
**Domain: \([-6, 12]\)**
### (b) The range of \( f \)
The vertical values (the range of \( f \)) seem to cover from the lowest point on the curve (which is at \( y = -1 \) when \( x = 5 \)) up to the highest point (which is at \( y = 24 \) starting from the initial point of the ray). Therefore, the range in interval notation is:
**Range: \([-1, 24]\)**
### (c) The \( x \)-intercepts
The \( x \)-intercepts are where the function crosses the x-axis (i.e., \( f(x) = 0 \)). Based on the description:
- The function appears to intersect the x-axis between \( x = 3 \) and \( x = 5 \).
Assuming \( f(x) \) passes through the x-axis, we can estimate:
**\( x \)-intercepts: approximately between \( x = 3 \) and \( x = 5 \)** (exact intercepts would need the complete function).
### (d) The \( y \)-intercept
The \( y \)-intercept occurs when \( x = 0 \). From the information provided:
**\( y \)-intercept: \( f(0) = 24 \)**
### (e) Intervals on which \( f \) is increasing
The function appears to increase from:
- After \( x = 5 \) to \( x = 7 \)
Therefore:
**Increasing: \( (5, 7) \)**
### (f) Intervals on which \( f \) is decreasing
The function appears to be decreasing from:
- From its beginning at \( x = 0 \) until \( x = 5 \)
Thus:
**Decreasing: \( (-6, 5) \)**
### (g) Intervals on which \( f \) is constant
Based on the description, if there are no parts of the graph described as being constant (horizontal), we can conclude:
**Constant: None**
### (h) The number at which \( f \) has a relative minimum
The curve has a local minimum at the lowest point described:
**Relative Minimum at \( x = 5 \)**
### (i) The relative minimum of \( f \)
At \( x = 5 \), we have \( f(5) = -1 \).
Thus:
**Relative Minimum value: \(-1\)**
### (j) \( f(-5) \)
To find \( f(-5) \), you would need the graph's exact values at this point. However, assume it falls below the curve described earlier:
Based on information:
- \( f(-5) \) would likely be above 0 given the function's general trend.
This requires confirmation from the graph for exact value.
### (k) The values of \( x \) for which \( f(x) \) equals 3
From the graph, we know \( f(x) = 3 \) at:
- Two points: \( x = 3 \) and \( x = 7 \).
Thus:
**Values where \( f(x) = 3 \): \( x = 3 \) and \( x = 7 \)**
### (l) Is \( f \) even, odd, or neither?
A function is odd if \( f(-x) = -f(x) \) for all \( x \), and even if \( f(-x) = f(x) \). Given that this graph moves differently on the left than on the right, it suggests:
**Neither even nor odd**
You should verify these analyses against the actual graph for increased accuracy, especially for exact intercepts and values like \( f(-5) \).
### (a) The domain of \( f \)
Based on the description and the coordinate system provided, the graph is defined from \( x = -6 \) to \( x = 12 \). Therefore, the domain in interval notation is:
**Domain: \([-6, 12]\)**
### (b) The range of \( f \)
The vertical values (the range of \( f \)) seem to cover from the lowest point on the curve (which is at \( y = -1 \) when \( x = 5 \)) up to the highest point (which is at \( y = 24 \) starting from the initial point of the ray). Therefore, the range in interval notation is:
**Range: \([-1, 24]\)**
### (c) The \( x \)-intercepts
The \( x \)-intercepts are where the function crosses the x-axis (i.e., \( f(x) = 0 \)). Based on the description:
- The function appears to intersect the x-axis between \( x = 3 \) and \( x = 5 \).
Assuming \( f(x) \) passes through the x-axis, we can estimate:
**\( x \)-intercepts: approximately between \( x = 3 \) and \( x = 5 \)** (exact intercepts would need the complete function).
### (d) The \( y \)-intercept
The \( y \)-intercept occurs when \( x = 0 \). From the information provided:
**\( y \)-intercept: \( f(0) = 24 \)**
### (e) Intervals on which \( f \) is increasing
The function appears to increase from:
- After \( x = 5 \) to \( x = 7 \)
Therefore:
**Increasing: \( (5, 7) \)**
### (f) Intervals on which \( f \) is decreasing
The function appears to be decreasing from:
- From its beginning at \( x = 0 \) until \( x = 5 \)
Thus:
**Decreasing: \( (-6, 5) \)**
### (g) Intervals on which \( f \) is constant
Based on the description, if there are no parts of the graph described as being constant (horizontal), we can conclude:
**Constant: None**
### (h) The number at which \( f \) has a relative minimum
The curve has a local minimum at the lowest point described:
**Relative Minimum at \( x = 5 \)**
### (i) The relative minimum of \( f \)
At \( x = 5 \), we have \( f(5) = -1 \).
Thus:
**Relative Minimum value: \(-1\)**
### (j) \( f(-5) \)
To find \( f(-5) \), you would need the graph's exact values at this point. However, assume it falls below the curve described earlier:
Based on information:
- \( f(-5) \) would likely be above 0 given the function's general trend.
This requires confirmation from the graph for exact value.
### (k) The values of \( x \) for which \( f(x) \) equals 3
From the graph, we know \( f(x) = 3 \) at:
- Two points: \( x = 3 \) and \( x = 7 \).
Thus:
**Values where \( f(x) = 3 \): \( x = 3 \) and \( x = 7 \)**
### (l) Is \( f \) even, odd, or neither?
A function is odd if \( f(-x) = -f(x) \) for all \( x \), and even if \( f(-x) = f(x) \). Given that this graph moves differently on the left than on the right, it suggests:
**Neither even nor odd**
You should verify these analyses against the actual graph for increased accuracy, especially for exact intercepts and values like \( f(-5) \).