Problem 134 - College Algebra with Modeling and Visualization

Question

If \(f(x)=a x^{2}+b x+c\) with \(a \neq 0,\) does \(f^{-1}(x)\) exist? Explain.

Step-by-Step Solution

Answer

No, \(f^{-1}(x)\) does not exist for a quadratic function like \(f(x) = ax^2 + bx + c\) because it is not one-to-one.

Step 1: Understand the Function Type

The given function \( f(x) = ax^2 + bx + c \) is a quadratic function because it is a polynomial of degree 2. Quadratic functions have a parabolic shape, and the standard form is \( y = ax^2 + bx + c \), where \( a eq 0 \).

Step 2: Determine the Nature of the Function

Quadratic functions are generally not one-to-one because they map more than one x-value to a single y-value. This is reflected in their symmetrical parabola shape, which makes it impossible for each y-value to correspond to only one x-value.

Step 3: Apply the Horizontal Line Test

To determine if a function has an inverse, you can use the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one, and thus the inverse does not exist. Since a parabola opens up or down and any horizontal line would intersect it twice, \( f(x) \) fails this test.

Step 4: Conclusion Based on Function Characteristics

Since \( f(x) = ax^2 + bx + c \) does not pass the horizontal line test and is not a one-to-one function, it does not have an inverse function. Only one-to-one functions can have inverses that are also functions.

Key Concepts

Quadratic Functions

Quadratic functions are a special type of polynomial function characterized by their second-degree term. A standard quadratic function takes the form:

• \( f(x) = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The defining feature of these functions is their graph, which is a parabola.
Parabolas can be either upward (if \(a > 0\)) or downward (if \(a < 0\)) facing. The symmetry of the parabola is determined by its axis of symmetry, given by the line \(x = -\frac{b}{2a}\).
Quadratic functions teach us about vertex points, critical points, and how these translate to real-world problems like projectile motion or optimizing areas within constraints. These functions' turning points, known as their vertices, can be calculated using the formula:

• Vertex, \( V(x, y) = (-\frac{b}{2a}, f(-\frac{b}{2a})) \)

One-to-One Functions

A function is termed one-to-one if each element in its range corresponds to exactly one element in its domain. This means that for each y-value, there is only one distinct x-value. Ensuring a function is one-to-one is crucial for finding inverses, as only one-to-one functions have inverses that are also functions.
To understand whether a function is one-to-one, we examine its graph. For a function to be one-to-one:

• No x-value can repeat different y-values.
• A y-value should originate from precisely one x-value.

Through understanding these concepts, students can appreciate the structure needed for a function to have an inverse. In essence, being one-to-one simplifies the process of reversing the mapping from inputs to outputs.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one, which is a prerequisite for the function to have an inverse. The test involves drawing horizontal lines across the graph of the function and observing their intersections.
The rules of the horizontal line test are:

• If every horizontal line drawn across the graph intersects it at most once, the function is one-to-one.
• If any horizontal line intersects the graph more than once, the function is not one-to-one.

For quadratic functions, which graph as parabolas, horizontal lines often intersect the graph more than once since these functions are symmetrical around the vertex. This symmetry means that multiple x-values map to the same y-value, leading them to fail the horizontal line test. Consequently, quadratic functions do not have inverses that are functions because they are not one-to-one.