Problem 22 - A Graphical Approach to College Algebra

Question

Decide whether each function is one-to-one. Do not use a calculator. $$y=\frac{1}{x+2}$$

Step-by-Step Solution

Answer

The function \( y = \frac{1}{x+2} \) is one-to-one.

Step 1: Definition of One-to-One Function

A function is considered one-to-one if every output value (y-value) is mapped from exactly one input value (x-value). In other words, no horizontal line should intersect the graph of the function at more than one point.

Step 2: Analyze the Given Function

The function provided is \( y = \frac{1}{x+2} \). This is a rational function that can be rewritten in terms of its transformation: it's essentially the reciprocal function \( \frac{1}{x} \), shifted 2 units to the left.

Step 3: Applying the Horizontal Line Test

To determine if the function is one-to-one, we apply the horizontal line test. A horizontal line will intersect the graph of \( y = \frac{1}{x+2} \) only once for any y-value, except at points where the function is undefined (such as \( x = -2 \)).

Step 4: Conclusion

Since no horizontal line intersects the graph at more than one point, the function \( y = \frac{1}{x+2} \) is one-to-one.

Key Concepts

Horizontal Line Test

To determine if a function is one-to-one, we apply the horizontal line test. This is a straightforward test. You imagine drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph only once, the function is one-to-one. This ensures each output is linked to a unique input.

In the provided exercise, we're checking the function \( y = \frac{1}{x+2} \). If you picture its graph, you'll notice that no horizontal line crosses it more than once, except where the function is undefined (like \( x = -2 \)). That means the function passes the horizontal line test! Therefore, we confirm that it is one-to-one.

Rational Function

A rational function is a type of function defined by the quotient of two polynomials. They generally take the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

The function \( y = \frac{1}{x+2} \) is a simple rational function where \( P(x) = 1 \) and \( Q(x) = x + 2 \). Rational functions often have specific characteristics, like vertical asymptotes and horizontal or oblique asymptotes.

For \( y = \frac{1}{x+2} \), there's a vertical asymptote at \( x = -2 \) because the denominator becomes zero here, making the function undefined. Understanding these traits helps identify the function's behavior across its domain.

Function Transformation

Function transformation involves shifting, stretching, compressing, or reflecting the graph of a function. By understanding these transformations, we can easily recognize how basic functions are modified.

For our example, \( y = \frac{1}{x+2} \) can be viewed as a transformed version of the basic reciprocal function \( y = \frac{1}{x} \). Here's how:

• Horizontal Shift: Adding 2 within the denominator \( (x + 2) \) shifts the graph 2 units to the left. This transformation changes the vertical asymptote from \( x = 0 \) to \( x = -2 \).

These transformations don't alter the one-to-one nature of the function but help in visualizing how the graph fits onto the coordinate plane. Understanding transformations is key in recognizing and sketching the graphs of various functions quickly.