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Chapter 7: Problem 14

Graph each relation and its inverse. $$ y=2 x-3 $$

Short Answer

Expert verified
The respective graphs of the original relation \(y = 2x - 3\) and its inverse \(y = (x + 3) / 2\) represent their linear equations. The graph of the inverse is the reflection of the original graph across the line \(y = x\).

Step by step solution

01

Reformulate the given equation

The given equation, \(y = 2x - 3\), should be rewritten by swapping \(x\) and \(y\). Therefore, it becomes \(x = 2y - 3\).
02

Find the inverse function

To find the inverse, resolve the reformulated equation for \(y\). Start by adding 3 to both sides of the equation: \(x + 3 = 2y\). Then, divide each side by 2: \((x + 3) / 2 = y\). Thus, the inverse function is \(y = (x + 3) / 2\).
03

Graph the original relation

Use the equation of the original relation, \(y = 2x - 3\), to plot its graph. Choose values for \(x\) and solve for \(y\). For instance, if \(x = 0\), then \(y = -3\); if \(x = 1\), then \(y = -1\). Use these points to draw the graph.
04

Graph the inverse relation

Use the equation of the inverse relation, \(y = (x + 3) / 2\), to plot its graph. Choose values for \(x\) and solve for \(y\). For instance, if \(x = 0\), then \(y = 1.5\); if \(x = 1\), then \(y = 2\). Use these points to draw the graph.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in mathematics. When you graph a linear equation, you're essentially plotting its solutions on a coordinate plane. A linear equation like \( y = 2x - 3 \) represents a straight line.
To draw this line, you can start by choosing a set of \( x \) values, then solving for \( y \). For example, for \( x = 0 \), \( y = -3 \); for \( x = 1 \), \( y = -1 \). These points are then plotted on the graph.
  • Find two points: The interception with the y-axis occurs when \( x = 0 \).
  • Slope: The coefficient of \( x \) (2 in our example) tells you the steepness and direction of the line. It means the graph rises 2 units for every 1 unit it moves to the right.
Once you have a few points plotted, you can draw a straight line through them to represent the equation.
Linear Relations
Linear relations are relationships between two quantities that change at a constant rate. In mathematical terms, this is represented by a linear function, which is essentially a straight line on a graph.
The equation \( y = 2x - 3 \) describes a linear function because throughout its graph, the ratio of change between \( y \) and \( x \) is constant. Here, for every unit increase in \( x \), \( y \) increases by 2 units. This characteristic constant rate of change is the hallmark of a linear relation.
  • Characteristics: Always straight lines when graphed, and have a constant slope.
  • Y-Intercept: This is where the line crosses the y-axis, and in \( y = 2x - 3 \), it's -3.
Linear relations can be easily identified by their equations, which have no exponents on the variables other than 1.
Solving Equations
Solving equations is a key process in algebra that allows us to find unknown values that make the equation true. For the equation \( y = 2x - 3 \), the unknowns provide a set of values that satisfy this equality.
Inverting functions changes x with y, leading to a new equation to solve for y. Begin with switching \( x \) and \( y \) in the equation \( x = 2y - 3 \). Solving means isolating \( y \) by similar arithmetic operations on both sides:
  • First, add 3 to each side: \( x + 3 = 2y \).
  • Next, divide everything by 2: \( \frac{x + 3}{2} = y \).
This method results in the inverse equation, allowing you to solve for any \( y \) corresponding to \( x \). Solving equations is an essential skill, as it builds on understanding relationships between variables.
Function Inversion
Function inversion is a mathematical process where the roles of the input and output of a function are reversed, essentially finding a function that 'undoes' another. This is especially useful in finding corresponding points on inverse functions.
For a function like \( y = 2x - 3 \), an inverse function is one where you swap \( x \) and \( y \), leading to \( x = 2y - 3 \), and solving for \( y \) gives \( y = \frac{x+3}{2} \).
  • Graphical Interpretation: The graph of an inverse is a reflection over the line \( y = x \).
  • Verification: Composing a function with its inverse should yield the original input values: \( f(f^{-1}(x)) = x \).
Understanding function inversion is crucial for analyzing situations where you need to revert a transformation or solve systems that naturally lend themselves to switching inputs and outputs efficiently.

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