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Chapter 6: Problem 3

By sketching a graph of \(y=3 x-1\) show that this is a one-to-one function.

Short Answer

Expert verified
Explain your reasoning based on the graph of the function. Answer: Yes, the function \(y = 3x - 1\) is a one-to-one function because it is a linear function with a non-zero slope. This means that each x-element in the domain is paired with a unique y-element in the codomain, ensuring there are no horizontal segments on the graph, and no two points have the same y-value.

Step by step solution

01

Understanding the function

The given function is \(y = 3x - 1\), which represents a linear function with a slope of 3 and a y-intercept at -1.
02

Determine the points to plot on the graph

It's helpful to choose a few points from the domain of the function to plot on the graph. For example, we can choose -1, 0, 1, and 2 as x-values, and then find the corresponding y-values using the function. For \(x = -1\), we have \(y = 3(-1) - 1 = -3 - 1 = -4\) For \(x = 0\), we have \(y = 3(0) - 1 = -1\) For \(x = 1\), we have \(y = 3(1) - 1 = 3 - 1 = 2\) For \(x = 2\), we have \(y = 3(2) - 1 = 6 - 1 = 5\)
03

Plot the points on the graph

Plot the points we found above on a graph: \((-1, -4)\), \((0, -1)\), \((1, 2)\), and \((2, 5)\).
04

Draw the line through the plotted points

Connect the plotted points with a straight line, as this is a linear function. This line should have an upward slope due to the positive coefficient (3) of the x-term.
05

Analyze the graph for one-to-one function

Observe the graph and check if any two points have the same y-value. Since the function is linear and has a non-zero slope, it will not have any horizontal segments, which means each x-element is paired with a unique y-element. Thus, the function is a one-to-one function.

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