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Chapter 3: Problem 15

Identify the set as a relation, a function, or both a relation and a function. $$ y=-6 x+7 $$

Short Answer

Expert verified
The set is both a relation and a function.

Step by step solution

01

- Understand the Given Equation

The equation provided is a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = -6 \) and \( b = 7 \).
02

- Define a Relation

A relation is a set of ordered pairs \( (x, y) \). Any equation that pairs an \( x \)-value with a \( y \)-value forms a relation.
03

- Determine if it is a Function

For the relation to be a function, each input \( x \) should have exactly one output \( y \). Since the equation \( y = -6x + 7 \) gives a unique \( y \)-value for each \( x \)-value, it meets the criteria of a function.
04

- Conclusion

Because the equation forms a set of ordered pairs and each \( x \)-value has a unique \( y \)-value, the set is both a relation and a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relations
In mathematics, a relation is a set of ordered pairs, like \((x, y)\). These pairs show how two elements, often numbers, are related to each other.
Each pair consists of one input and its corresponding output. Relations can be represented in many forms, such as:
  • Equations
  • Graphs
  • Tables
A key point to remember is that every equation involving variables creates a relation. For example, the equation given, \(-6x + 7\), each value of \`x\` generates a unique value of \`y\`, forming multiple ordered pairs like \((-1, 13), (0, 7), (1, 1)\), and so on. Understanding relations is the first step toward grasping more complex mathematical concepts.
Functions
A function is a specific type of relation. For a relation to qualify as a function, each input \`x\` must correspond to exactly one output \`y\`. In simpler terms, no \`x\`-value can map to multiple \`y\`-values.
How do you determine if a relation is a function? Two key methods are:
  • Vertical Line Test: Draw vertical lines across the graph of the relation. If any line crosses the graph more than once, it's not a function.
  • Analytical Approach: Check the equation to see if you get one value of \`y\` for every \`x\`.
For the equation \(-6x + 7\), each input value of \`x\` will always produce one unique output for \`y\`. Thus, it is both a relation and a function.
Slope-Intercept Form
The slope-intercept form of a linear equation is \(`y = mx + b`\). This form makes it easy to graph a line and understand its behavior.
Here, \`m\` represents the slope, and \`b\` represents the y-intercept of the line.
Let's break it down:
  • Slope (m): It tells us how steep the line is and the direction it goes. In our equation, \`m = -6\`, meaning the line falls 6 units for every 1 unit it moves to the right.
  • Y-Intercept (b): This is where the line crosses the y-axis. For \`b = 7\`, it tells us the line crosses the y-axis at (0, 7).
Using this form, you can graph the equation, identify the slope and y-intercept, and understand the relationship between \`x\` and \`y\`. The simplicity and clarity of the slope-intercept form make it fundamental for studying linear equations.

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Most popular questions from this chapter

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of fresh tomatoes consumed per person in the United States increased from \(89.9\) lb in 2009 to 93.5 lb in 2011. (Source: www.ers.usda.gov, Dec. 15,2011\()\)

Use the slope formula to find the slope of the line that passes through the points. \((-20,45) ;(-15,90)\)

For exercises 67-78, (a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(4 x+9 y=72\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-5) ;(2,0)\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of traffic fatalities in Missouri decreased from 1257 deaths in 2005 to 819 deaths in 2010. Round to the nearest whole number. (Source: www- nrd.nhtsa .dot.gov)
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