Problem 25 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=-2 x^{2}+x $$

Step-by-Step Solution

Answer

The function is not one-to-one because it fails the horizontal line test.

Step 1: Understanding the Function

The function given is \( f(x) = -2x^2 + x \). This is a quadratic function, which typically forms a parabola when plotted on a graph.

Step 2: Determine the Graph's Shape

Since the coefficient of \( x^2 \) is negative (-2), the parabola opens downwards. It has a vertex that acts as a maximum point.

Step 3: Apply the Horizontal Line Test

A function is one-to-one if and only if every horizontal line crosses the graph of the function at most once. Given the downward-opening parabola, most horizontal lines will intersect the graph at two points.

Step 4: Conclusion from the Test

Since there are horizontal lines that intersect the parabola more than once, the function \( f(x) = -2x^2 + x \) is not one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple yet powerful tool for determining whether a function is one-to-one. The idea is straightforward: if you can draw any horizontal line that crosses the graph of the function more than once, then the function is not one-to-one. This is because a one-to-one function, sometimes called an injective function, maps each input to a unique output.

In our example, we're dealing with the function \( f(x) = -2x^2 + x \), which forms a downward-opening parabola. When we apply the horizontal line test, we find that many horizontal lines intersect this parabola at two different points. Thus, the function cannot be one-to-one, because multiple inputs are mapped to the same output. Overall, the horizontal line test gives us a clear visual way to assess the nature of the function.

Quadratic Function

Quadratic functions are a type of polynomial function represented by the standard form \( ax^2 + bx + c \). In our example, the quadratic function is \( f(x) = -2x^2 + x \). The coefficient of \( x^2 \) is critical in determining the graph's direction. If the coefficient is positive, the parabola opens upwards, and if it's negative, like in our function, it opens downwards.

Every quadratic function has a characteristic U-shaped curve known as a parabola. The vertex of the parabola is the highest or lowest point on the graph, depending on the direction it opens. This function's vertex represents the point where the output value is maximized since it opens downward. Understanding the behavior of quadratic formulas is crucial for spotting patterns and predicting graph shapes.

Graphing Functions

Graphing functions can often illuminate their properties and behavior far better than numerical calculations alone. When graphing \( f(x) = -2x^2 + x \), begin by identifying key components such as the vertex and axis of symmetry. The vertex can be found using the formula \(-\frac{b}{2a}\) which gives you the x-coordinate of the vertex. For our function, this position is where the parabola reaches its maximum height.

Once you've marked the vertex, you can determine how the rest of the parabola behaves by plotting some additional points. This step illustrates the parabola's symmetrical nature. Graphs not only aid in visualizing mathematical concepts but also in applying tests like the horizontal line test. Adding visual elements to your learning routine can aid in better comprehension of how quadratic functions perform across their domain.