Problem 4 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The function is not one-to-one.

Step 1: Determine the Definition of a One-to-One Function

A function is considered one-to-one (injective) if and only if \(f(a) = f(b)\) implies \(a = b\) for every \(a, b \) in the function's domain. This means that no two different inputs should map to the same output for a function to be one-to-one.

Step 2: Analyze the Function Structure

Examine the function \(f(x) = -x^2\). This function is a basic transformation of the quadratic function \(x^2\) where each output is negated. Quadratic functions typically produce outputs that are not one-to-one due to their symmetrical nature (i.e., both positive and negative values for \(x\) give the same output). For example, \(f(1) = -1\) and \(f(-1) = -1\).

Step 3: Use the Horizontal Line Test

Graph \(f(x) = -x^2\) or visualize it. The graph of \(f(x)\) is a parabola that opens downwards. According to the horizontal line test, a function is not one-to-one if a horizontal line intersects its graph more than once. Clearly, any horizontal line below the vertex of the parabola will intersect the graph twice.

Step 4: State the Conclusion

Since the function \(f(x) = -x^2\) fails the horizontal line test, it is not one-to-one. Thus, from both theoretical analysis and graphical representation, it is clear that the condition for a function being one-to-one is not satisfied.

Key Concepts

Function Analysis

Function analysis is a key step in understanding the behavior and characteristics of a function. A function is a relation between a set of inputs and a set of possible outputs. To determine if a function is one-to-one, we need to assess whether distinct inputs produce distinct outputs. For the function in the exercise, \( f(x) = -x^2 \), function analysis reveals it as a transformation of the basic quadratic function \( x^2 \). Here, each output value is negated. When examining one-to-one functions:

• Each output value must be unique for each input value.
• If the same output can be mapped from two different inputs, the function is not one-to-one.

In this case, the value of \( f(x) \) for positive and negative values of \( x \) could be the same. Therefore, understanding the core concept of a one-to-one function helps to quickly identify that \( f(x) = -x^2 \) does not meet this criterion.

Horizontal Line Test

The horizontal line test is a simple yet powerful visual tool used to determine whether a function is one-to-one. To apply this test, we imagine drawing horizontal lines across the graph of the function in various places. For \( f(x) = -x^2 \), the graph forms a downward-opening parabola. When conducting the horizontal line test:

• If a horizontal line crosses the graph more than once, then the function is not one-to-one.
• If every horizontal line crosses the graph at most once, the function is one-to-one.

In this exercise, any horizontal line occurring below the apex (or vertex) of the parabola will intersect the curve twice. This confirms that \( f(x) = -x^2 \) is not one-to-one. The horizontal line test is a quick and effective way to visually identify one-to-one functions.

Quadratic Functions

Quadratic functions play a fundamental role in mathematics, characterized by the equation \( ax^2 + bx + c \). These functions graph as U-shaped curves called parabolas. For \( f(x) = -x^2 \), it's a simple quadratic function with a downward opening.Key properties of quadratic functions include:

• Vertex: The highest or lowest point on the graph, where the direction changes.
• Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves.
• Not One-to-One: Quadratic functions are generally not one-to-one due to their symmetry, unless restricted to a specific domain (like only non-negative \( x \) values).

In the example \( f(x) = -x^2 \), the symmetry means that positive and negative values of \( x \) yield the same output, which further confirms that it is not a one-to-one function. Understanding these characteristics is vital for analyzing quadratic functions and determining their injectivity.