Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16
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2. Graphs of Equations
Graphs and Coordinates
2:10 minutesProblem 25
Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. |x| − y = 2
Verified step by step guidance1
Rewrite the given equation to isolate \( y \) on one side. Starting with \( |x| - y = 2 \), subtract \( |x| \) from both sides to get \( -y = 2 - |x| \).
Multiply both sides of the equation by \( -1 \) to solve for \( y \), resulting in \( y = |x| - 2 \).
Recall the definition of a function: for each input \( x \), there must be exactly one output \( y \).
Since \( y = |x| - 2 \) expresses \( y \) explicitly in terms of \( x \) and the absolute value function \( |x| \) produces a unique output for each \( x \), this equation defines \( y \) as a function of \( x \).
Therefore, the equation \( |x| - y = 2 \) defines \( y \) as a function of \( x \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Function
A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we check if for every x there is only one y. If any x maps to multiple y-values, the relation is not a function.
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Solving for y in Terms of x
To analyze whether y is a function of x, isolate y on one side of the equation. This helps identify if y can be expressed as a single-valued expression in terms of x. If solving yields multiple y-values for a single x, y is not a function of x.
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Absolute Value Properties
The absolute value |x| represents the distance of x from zero and is always non-negative. Understanding how |x| behaves is important when manipulating equations involving absolute values, as it affects the domain and range and can influence whether y is uniquely defined.
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