Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=-(x+3)^{2}-8$$

Step-by-Step Solution

Answer

The function is not one-to-one because it is a downward-opening quadratic.

Step 1: Understanding One-to-One Functions

A function is considered one-to-one if every element of the range is mapped to by exactly one element of the domain. This means that if \( f(a) = f(b) \), then \( a = b \) must be true for the function to be one-to-one.

Step 2: Analyzing the Given Function

The given function is \( y = -(x+3)^2 - 8 \). This is a quadratic function written in vertex form \( y = a(x-h)^2 + k \) with \( a = -1 \), \( h = -3 \), and \( k = -8 \). Since \( a < 0 \), the parabola opens downwards.

Step 3: Testing for One-to-One Property

For a function to be one-to-one, each \( y \)-value should be unique for each \( x \)-value within its domain. Since the function is a downward-opening quadratic, it fails the horizontal line test (a line can intersect the parabola at more than one point), meaning it is not one-to-one.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function that are characterized by the highest degree term being squared. Their general form is given by \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These functions graph into a U-shaped curve known as a parabola. Here are some key features of quadratic functions:

• Vertex: The point where the parabola changes direction. For upward opening parabolas, it's the minimum point, and for downward opening ones, it's the maximum point.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images.
• Direction of Opening: Determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Quadratic functions often model real-world scenarios such as projectile motion and areas, due to their symmetric properties.

Vertex Form

The vertex form of a quadratic function is a special representation that makes it easy to identify the vertex of the parabola. It is written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For our function, \( y = -(x+3)^2 - 8 \), the vertex is located at \((-3, -8)\). Understanding vertex form is crucial as it allows for a quick read of the parabola's key characteristics:

• Vertex: Easily identified as \((h, k)\) in the expression.
• Axis of Symmetry: The line \( x = h \) goes through the vertex and divides the parabola symmetrically.
• Direction: Determined by \( a \). Here, \( a = -1 \), so the parabola opens downward, indicating \( h \) is a maximum point.

The vertex form makes graph transformations easy to manage and vividly visualizes translations and reflections.

Horizontal Line Test

The horizontal line test is a handy tool to determine if a function is one-to-one. A function is considered one-to-one if any horizontal line crosses its graph at most once. This test is essential when considering inverse functions since only one-to-one functions have inverses. In the context of quadratic functions, particularly our function \( y = -(x+3)^2 - 8 \), applying the horizontal line test reveals that the function is not one-to-one. A horizontal line can intersect the downward-opening parabola in more than one location, confirming that multiple \( x \)-values can result in the same \( y \)-value. In functions like this quadratic, the repeated intersection indicates many-to-one mapping within its domain, meaning not every output is unique. Thus, without restricting its domain, this particular quadratic cannot have an inverse function.