2. Graphs of Equations

Graphs and Coordinates

2. Graphs of Equations

Graphs and Coordinates

2:10 minutes

Problem 26a

Textbook Question

Determine whether each equation defines y as a function of x. |x|- y = 5

Verified step by step guidance

Step 1: Recall the definition of a function. A function is a relation where each input (x) corresponds to exactly one output (y). To determine if the given equation defines y as a function of x, we need to check if y is uniquely determined for every value of x.

Step 2: Start with the given equation: |x| - y = 5. Rearrange it to isolate y. Subtract |x| from both sides: -y = 5 - |x|.

Step 3: Multiply through by -1 to solve for y: y = |x| - 5. This equation expresses y explicitly in terms of x.

Step 4: Analyze the equation y = |x| - 5. The absolute value function |x| is defined for all real numbers x and always produces a single value. Subtracting 5 from |x| does not introduce ambiguity, so y is uniquely determined for every x.

Step 5: Conclude that the equation y = |x| - 5 defines y as a function of x because each x corresponds to exactly one y value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Definition

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 must check if for every x, there is a unique y. This is often visualized using the vertical line test, where a vertical line intersects the graph of the relation at most once.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In the equation |x| - y = 5, the absolute value introduces two cases for y depending on whether x is positive or negative. Understanding how absolute values behave is crucial for analyzing the equation and determining the nature of the relationship between x and y.

Rearranging Equations

Rearranging an equation involves manipulating it to isolate one variable in terms of the other. In this case, we can express y in terms of x by rewriting the equation |x| - y = 5 as y = |x| - 5. This step is essential for evaluating whether y is a function of x, as it allows us to directly assess the uniqueness of y for each x.

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In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

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