Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=(x-2)^{2}$$

Step-by-Step Solution

Answer

The function \(y = (x-2)^2\) is not one-to-one.

Step 1: Understand the Concept of One-to-One Function

A function is considered one-to-one if for every distinct pair of input values \(a\) and \(b\), the corresponding output values \(f(a)\) and \(f(b)\) are also distinct. This means that no horizontal line should intersect the graph of the function more than once.

Step 2: Analyze the Given Function

The given function is \(y = (x-2)^2\). This is a quadratic function with its vertex shifted to the point \((2, 0)\). Quadratic functions generally have a symmetric parabolic shape centered around their vertex.

Step 3: Check for One-to-One Property Using Horizontal Line Test

To determine if \(y = (x-2)^2\) is one-to-one, use the horizontal line test: draw horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function is not one-to-one. The graph of \(y = (x-2)^2\) is a parabola that opens upwards, and any horizontal line above the vertex \((2, 0)\) will intersect the graph twice, once on each side of the vertex.

Step 4: Conclusion on One-to-One Property

Since the horizontal line test shows that the graph of \(y = (x-2)^2\) is intersected at two points in various places, it indicates that the function is not one-to-one. This occurs because multiple input values (specifically \(xeq2\)) can produce the same output.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple yet effective method to determine if a function is one-to-one. A function is identified as one-to-one if no horizontal line intersects its graph at more than one point. This is crucial because it ensures that each output value of the function corresponds to exactly one input value.

To apply the test, you imagine drawing horizontal lines across the graph of the function. If any line crosses the graph in more than one spot, the function is not one-to-one. This typically happens for graph shapes like circles or parabolas. For example, with the function given in the exercise, you can visualize these horizontal lines crossing the parabola. It's clear they will intersect the curve twice above the vertex. This confirms that it is not one-to-one.

• Use for determining if a function is one-to-one
• Involves drawing horizontal lines across the function's graph
• If any line crosses more than once, the function fails the test

Quadratic Functions

Quadratic functions are common in mathematics and are typically expressed in the form of \(y = ax^2 + bx + c\). Their graphs are called parabolas, which have a characteristic U shape. Depending on the sign of the coefficient \(a\), the parabola can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

The given function is \(y = (x-2)^2\). This is a specific type of quadratic function, where the parabola is shifted to the point (2, 0). This shift moves the vertex of the parabola away from the origin. The graph has a symmetric shape about this vertex, which plays a crucial role in determining its one-to-one property.

• Expressed in the standard form \(y = ax^2 + bx + c\)
• Graph is a parabola, U-shaped curve
• Vertex indicates the parabola's maximum or minimum

Function Analysis

Analyzing functions involves understanding their properties and behaviors. This includes determining the type of function, its graph shape, and key features such as symmetry, intercepts, and one-to-one nature.

For function \(y = (x-2)^2\), function analysis reveals that it is a quadratic function with a symmetrical parabola centered at its vertex, \((2, 0)\). The symmetry about the vertex indicates that the graph does not satisfy the horizontal line test; thus, it is not a one-to-one function. Recognizing these patterns helps in various mathematical tasks, like solving equations and graph interpretation.

• Involves studying graph shapes, symmetry, and key features
• Aims to determine important properties of functions
• Crucial for tasks like solving equations and interpreting graphs