Problem 10 - Essential Calculus Early Transcendentals

Question

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ f(x)=10-3 x $$

Step-by-Step Solution

Answer

Yes, the function is one-to-one.

Step 1: Understanding One-to-One Function

A function is considered one-to-one if it never assigns the same value to two different inputs. In other words, for it to be one-to-one, if \( f(x_1) = f(x_2) \), then it must be that \( x_1 = x_2 \).

Step 2: Analysis Using Algebraic Method

Consider the function \( f(x) = 10 - 3x \). To determine if it's one-to-one, assume \( f(x_1) = f(x_2) \). This implies \( 10 - 3x_1 = 10 - 3x_2 \). Simplifying, we get \(-3x_1 = -3x_2\) or further simplified, \( x_1 = x_2 \). This shows that the function is one-to-one because the equality holds only when \( x_1 = x_2 \).

Step 3: Graphical Interpretation

The function \( f(x) = 10 - 3x \) is a linear function, represented graphically as a straight line with a negative slope. A linear function with a non-zero slope is always one-to-one because it passes the horizontal line test: no horizontal line intersects the graph more than once.

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed. It can be represented by the equation of the form \( f(x) = mx + b \) where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept

The slope, \( m \), determines the steepness and direction of the line. If \( m \) is positive, the line inclines upwards, and if negative, the line declines.

Linear functions are simple because they have constant rates of change. This means for any equal changes in \( x \), the changes in \( f(x) \) will also be equal. In our particular function, \( f(x) = 10 - 3x \), the "-3" indicates the line has a negative slope, meaning it falls as \( x \) increases.

Horizontal Line Test

The horizontal line test is a simple way to determine if a function is one-to-one. By applying the test to a graph, you draw horizontal lines across the graph and check how many times each line intersects with the graph.

• If any horizontal line crosses the graph more than once, the function is not one-to-one.
• If every horizontal line crosses the graph at most once, then the graph represents a one-to-one function.

For the linear function \( f(x) = 10 - 3x \), a graph of this function will show a sloped line. This line will pass the horizontal line test, confirming it is one-to-one because any horizontal line will intersect it just once.

Function Analysis

When performing function analysis to determine if a function is one-to-one, you examine its properties either graphically or algebraically. For algebraic analysis, if given \( f(x_1) = f(x_2) \), you solve to find whether \( x_1 = x_2 \). If they are always equal, it means the function is one-to-one.

In the given example, starting with \( f(x_1) = f(x_2) \) for \( f(x) = 10 - 3x \), simplifying reveals \(-3x_1 = -3x_2\), thus \( x_1 = x_2 \). Therefore, using algebraic reasoning, we confirm this function's one-to-one property.

Function Graph

A function graph visually represents the relationship between input values (x-axis) and output values (y-axis). In the case of the linear function \( f(x) = 10 - 3x \), the graph is a straight line.

To graph a linear function, you can use:

• The y-intercept \( b \) as the starting point on the y-axis which is 10 in this case.
• The slope \( m \) (which is -3 here) to determine the next points. From the y-intercept, for each step of 1 unit move horizontally, move 3 units downward because the slope is negative.

By plotting these points and drawing a line through them, the graph can quickly reveal if the function is one-to-one. For \( f(x) = 10 - 3x \), the graph will confirm the function's slope ensures it passes the horizontal line test.