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Chapter 4: Problem 13

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=3 x\)

Short Answer

Expert verified
The graph of \(y = 3x\) is a straight line through the origin. It is a direct variation and a one-to-one function.

Step by step solution

01

Identify the Function Type

The function given is \( y = 3x \). This is a linear function where the coefficient of \(x\) is 3. A linear function can generally be written in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. Here, \( m = 3 \) and \( b = 0 \).
02

Graph the Function

To graph the function \( y = 3x \), identify the slope \( m = 3 \). This means for every 1 unit increase in \( x \), \( y \) increases by 3 units. The y-intercept is at \( b = 0 \), so the graph passes through the origin (0,0). Plot points by choosing values for \( x \) (e.g., \( x = 1, 2, -1 \) ) and compute corresponding \( y \) values (e.g., \( y = 3 \times 1 = 3 \), \( y = 3 \times 2 = 6 \), \( y = 3 \times (-1) = -3 \)). Draw a straight line through these points.
03

Check for Direct Variation

A function is a direct variation if it can be expressed as \( y = kx \) where \( k \) is the constant of variation. Here, \( y = 3x \), which matches the direct variation form with \( k = 3 \). Therefore, this function is a direct variation.
04

Determine if the Function is One-to-One

A function is one-to-one if every \( y \) value has exactly one corresponding \( x \) value. For the function \( y = 3x \), each \( x \) produces a unique \( y \) because multiplying any input \( x \) by 3 will give a unique output. Therefore, the function is one-to-one.

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Key Concepts

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

Graphing Functions
Graphing linear functions is a fundamental skill in understanding algebra. It involves plotting points on a coordinate plane. For the function \( y = 3x \), the process of graphing involves understanding the slope and intercept:
  • **Slope (\( m \)):** This is a measure of how steep the line is. For \( y = 3x \), the slope is 3, indicating that for every 1 unit increase in \( x \), \( y \) increases by 3 units.
  • **Y-intercept (\( b \)):** This is the point where the line crosses the y-axis. In \( y = 3x \), the y-intercept is 0, meaning the line passes through the origin \((0,0)\).
To graph \( y = 3x \):
  • Start at the origin \((0,0)\), which is also the y-intercept.
  • From the origin, move up 3 units (the rise) and 1 unit to the right (the run) to plot the next point \((1,3)\).
  • Continue plotting more points, like \((2,6)\) and \((-1,-3)\), following the same rise/run pattern.
Connect these points with a straight line, extending across the plane. This visual representation helps in understanding how the function behaves overall.
Direct Variation
Direct variation is a special type of linear function where the relationship between two variables can be described by the equation \( y = kx \). Here, \( k \) is the constant of variation, and it represents the rate at which \( y \) changes with respect to \( x \). In our given function \( y = 3x \):
  • The equation matches the direct variation format \( y = kx \), with \( k = 3 \).
  • This means as \( x \) increases or decreases, \( y \) does the same in direct proportion.
Direct variation implies:
  • Passing through the origin (\((0,0)\) point), because when \( x = 0 \), \( y \) must be 0. This is true for \( y = 3x \).
  • The line will always have a constant slope, illustrated by the ratio \( \frac{y}{x} = k \).
Understanding direct variation is useful for modeling real-world situations where one quantity changes at a constant rate with another.
One-to-One Functions
One-to-one functions, also known as injective functions, have the important property that each \( x \) value has a unique \( y \) output, and each \( y \) value is paired with exactly one \( x \) value. This characteristic is essential for functions that can be reversed (i.e., have an inverse function):
  • To confirm if a function is one-to-one, use the **Horizontal Line Test**. If any horizontal line intersects the graph of the function at most once, the function is one-to-one.
  • For our function \( y = 3x \), applying this test shows that each \( y \) value corresponds to one unique \( x \) value. Hence, it is one-to-one.
  • This property implies that no two different \( x \) values produce the same \( y \) value.
One-to-one functions are particularly important when solving equations and working with function inverses, where reversing the process should lead back to the original value.

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Most popular questions from this chapter

Explain the difference between direct variation and inverse variation.

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \left\\{(x, y) : y=x^{2}+2 \text { for } 0 \leq x \leq 5\right\\} $$

In \(17-20 :\) a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers. $$ g(x)=x-5 $$

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{f}^{-1}(\mathrm{f}(-6)) $$

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(0,8),(1,7),(2,6),(3,5),(4,4)\\} $$
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