Problem 19 - Elementary and Intermediate Algebra

Question

Determine whether each function is one-to-one. \(f(x)=2 x\)

Step-by-Step Solution

Answer

The function \(f(x) = 2x\) is one-to-one because each input has a unique output.

Step 1: Understanding One-to-One Functions

A function is considered one-to-one if each value in the domain maps to a unique value in the range. This means no two different inputs (x-values) will produce the same output (y-value).

Step 2: Examining the Linear Function

The given function is linear and given by the equation \(f(x) = 2x\). A linear function of form \(f(x) = ax + b\) is one-to-one if \(a eq 0\). In this case, \(a = 2\), which is not zero.

Step 3: Using the Horizontal Line Test

The horizontal line test states that a function is one-to-one if no horizontal line intersects the graph more than once. For \(f(x) = 2x\), any horizontal line will intersect the graph at most once.

Step 4: Testing with Algebra

To prove one-to-oneness algebraically, assume \(f(x_1) = f(x_2)\). For \(f(x) = 2x\), this means: \[ 2x_1 = 2x_2 \] Dividing both sides by 2, we get:\[ x_1 = x_2 \] This shows that each output is produced by exactly one input.

Key Concepts

Linear Functions

Linear functions are among the simplest types of functions you encounter in mathematics. They have the general form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. Linear functions create straight lines when graphed. The parameter \( a \) represents the slope of the line, dictating how steep the line is. Meanwhile, \( b \) is the y-intercept, indicating where the line crosses the y-axis.
In our exercise, the function given was \( f(x) = 2x \), a linear function with \( a = 2 \) and \( b = 0 \). This means the line passes through the origin (0,0) and has a slope of 2, climbing upwards to the right.
The property that determines whether a linear function is one-to-one is its slope. As long as \( a \), the slope, is not zero, the linear function is one-to-one. In our case, since \( a = 2 \), the line is non-horizontal and differs in output for every different input. This confirmed the one-to-one nature of the function.

Horizontal Line Test

The horizontal line test is a visual tool used to determine if a function is one-to-one. If you can draw a horizontal line through the graph of a function and it touches the graph at most once, the function passes the test and is one-to-one.
For the function \( f(x) = 2x \), any horizontal line you draw will only intersect the graph once. Why? Because the graph of \( f(x) = 2x \) is a straight, diagonal line through the origin, constantly moving upwards as you move right.

• This inclination ensures that each horizontal level crosses the line in only one place.
• It's crucial for functions to pass this test since it confirms each input has a unique output, a defining trait of one-to-one functions.

Therefore, the horizontal line test is a quick check to visually assure that no two different input values will share the same output in a one-to-one function.

Function Domain and Range

When considering functions, knowing the domain and range is essential. The domain of a function is the complete set of possible input values (often \( x \)-values), while the range is the set of possible outputs (\( y \)-values).
For linear functions like \( f(x) = 2x \), the domain is all real numbers because you can input any real number for \( x \). Similarly, the range is also all real numbers because as \( x \) takes on any real value, \( 2x \) produces any real number output.
One-to-one functions like \( f(x) = 2x \) ensure every element of the range is paired with exactly one element from the domain. Hence:

• The domain \( (-\infty, \infty) \) means you can plug any x value.
• The range \( (-\infty, \infty) \) means your outputs cover all real numbers.

Understanding these concepts helps reinforce the one-to-one nature of the function, since it emphasizes the idea of unique mapping from inputs to outputs, solidifying the understanding of its behavior over its entire graph.