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Magazine Chapter 2: Problem 21
Graph each function. Identify the domain and range. \(h(x)=|-x|\)
Short Answer
Expert verified
The graph is a V-shaped curve symmetric around the y-axis. Domain: (-鈭�, 鈭�); Range: [0, 鈭�)
Step by step solution
01
Understand Absolute Value
Understand that the absolute value function, denoted by \(|x|\), measures the distance of a number from zero on the number line. It always outputs a non-negative value. In this case, \(-x\) becomes positive when the absolute value is taken, making \(|-x| = |x|\).
02
Identify Parent Function
The parent function for \(|-x|\) is \(|x|\), which is a V-shaped graph that points upwards, symmetry along the y-axis.
03
Consider Transformations
Since \(|-x|\) turns negative x-values into positive ones inside the function, it does not alter the graph's orientation or shape compared to \(|x|\). \(|-x|\) will mirror itself with respect to the y-axis, but the function remains unchanged visually as \(|x|\).
04
Graph the Function
Plot the absolute value function: starting from the origin (0,0), it forms a V-shape with lines y=x for x鈮�0 and y=-x for x<0. The graph opens upwards and remains symmetric about the y-axis.
05
Determine the Domain
The function \(|-x|\) accepts all real numbers as input, since you can take the absolute value of any real number. Therefore, the domain is all real numbers, \((-fty, fty)\).
06
Determine the Range
The output of \(|-x|\) is always non-negative, meaning the lowest value it can take is 0 and it can extend to positive infinity. Thus, the range is \[0, fty)\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain and Range
The domain and range are important characteristics of a function. Understanding these can help you identify the limitations and extent of the function.
- Domain: This represents all possible input values (x-values) that can be fed into the function. For the absolute value function \(|-x|\), since you can take the absolute value of any real number, the domain is all real numbers. In mathematical terms, it is expressed as \((-fty, fty)\).
- Range: This pertains to all possible output values (y-values) produced by the function. Since the absolute value produces non-negative results, the smallest y-value achievable by \(|-x|\) is 0, extending to positive infinity. Thus, the range for this function is \([0, fty)\).
Parent Function
The concept of a parent function is crucial as it helps in understanding transformations and variations of functions. A parent function represents the simplest form of a class of functions.
In this scenario, the parent function of \(|-x|\) is \(|x|\), an absolute value function. This parent function is characterized by a V-shaped graph with a distinct vertex at the origin \((0, 0)\). It typically opens upwards and has an axis of symmetry along the y-axis.
Recognizing parent functions can simplify graphing complex functions, as it serves as a reference point for shifts, stretches, or reflections.
In this scenario, the parent function of \(|-x|\) is \(|x|\), an absolute value function. This parent function is characterized by a V-shaped graph with a distinct vertex at the origin \((0, 0)\). It typically opens upwards and has an axis of symmetry along the y-axis.
Recognizing parent functions can simplify graphing complex functions, as it serves as a reference point for shifts, stretches, or reflections.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize a function's behavior.
For the function \(h(x) = |-x|\), the graph is identical to that of the parent function \(|x|\). It forms a V-shape starting at the origin \((0, 0)\).
For the function \(h(x) = |-x|\), the graph is identical to that of the parent function \(|x|\). It forms a V-shape starting at the origin \((0, 0)\).
- To graph, begin by plotting key points: \((0, 0)\), \((1, 1)\), \((-1, 1)\)
- Draw lines through these points extending towards infinite points in both upper quadrants.
Symmetry
Symmetry is a feature that simplifies understanding of a function's behavior. A function is symmetric if a portion of the graph mirrors another part of the graph.
The function \(h(x) = |-x|\) showcases symmetry across the y-axis. This means the left half of the graph is a mirror image of the right half.
The function \(h(x) = |-x|\) showcases symmetry across the y-axis. This means the left half of the graph is a mirror image of the right half.
- This particular y-axis symmetry is noted in the parent function \(|x|\) as well.
- Even when negative x-values are considered, \(|-x|\ = |x|\), demonstrating that the graph's shape and orientation remain unaffected.
Transformations
Transformations involve changing a function's position or shape while retaining its core identity.
The function \(h(x) = |-x|\) results from reflecting the x-values in the absolute value operation.
The function \(h(x) = |-x|\) results from reflecting the x-values in the absolute value operation.
- Reflections: For \(|-x|\), the negative input values are mirrored to positive, creating a reflection over the y-axes. However, due to the inherent property of the absolute value, \(|-x|\ = |x|\), the graph remains identical to \(|x|\).
- No translation, stretching, or compressing occurs for \(h(x)\), making it straightforward to recognize and graph from the parent function.
