Problem 24 - 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. $$ f(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

The function \( f(x) = x^2 - 1 \) is not one-to-one because it fails the horizontal line test.

Step 1: Understanding the Function

The function given is \( f(x) = x^2 - 1 \). This is a quadratic function, known to generally produce a parabolic curve when graphed. The parabola opens upwards because the coefficient of \( x^2 \) is positive.

Step 2: Graphing the Function

To determine if \( f(x) \) is one-to-one, we can graph the function. The vertex of the parabola is at (0, -1), and it is symmetric about the y-axis. The basic shape is a standard parabola shifted 1 unit downward.

Step 3: Applying the Horizontal Line Test

A function is one-to-one if no horizontal line intersects the graph at more than one point. For the graph of \( f(x) = x^2 - 1 \), any horizontal line drawn in the range \( y > -1 \) will intersect the parabola at two points.

Step 4: Conclusion on One-to-One Property

Since horizontal lines can intersect the graph of \( y = x^2 - 1 \) at two points, the function fails the horizontal line test. Therefore, \( f(x) = x^2 - 1 \) is not one-to-one.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a graphical method used to determine if a function is one-to-one. A function is considered one-to-one if, for every horizontal line drawn through its graph, the line intersects the graph at most once. This implies that for every output value, there is only one input value.

• If a horizontal line intersects the graph of the function more than once, the function is not one-to-one.
• This test is especially useful for visual assessments when dealing with polynomial functions.

For the function in question, \( f(x) = x^2 - 1 \), graphing it shows that possible horizontal lines can intersect the parabola at up to two points. Therefore, we conclude that \( f(x) \) is not one-to-one.

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is two. It is generally represented in the form \( f(x) = ax^2 + bx + c \). The essential properties of a quadratic function include:

• The graph is a parabola, which can open either upwards or downwards depending on the sign of \( a \).
• The vertex of the parabola represents its maximum or minimum point.
• It is symmetric about a vertical line through the vertex, known as the axis of symmetry.

In the equation \( f(x) = x^2 - 1 \), the graph is a parabola opening upwards with its vertex at the point \((0, -1)\). This symmetry about the y-axis results from the absence of a linear \( x \) term. Due to this symmetrical property, the function is not one-to-one, as tested by the horizontal line test.

Graphing Functions

Graphing functions is a crucial skill for visualizing relationships between variables and understanding the properties of a function. The process often involves:

• Identifying the type of function you have, such as linear, quadratic, or exponential.
• Plotting key points, such as intercepts and the vertex, especially for quadratic functions.
• Observing the overall shape and direction of the graph to understand behavior such as increasing, decreasing, or constant parts.

When graphing \( f(x) = x^2 - 1 \), you start by plotting the vertex at \((0, -1)\), then choose additional points for an accurate parabola shape. Monitoring the graph can reveal much, including symmetry, direction of opening, and the nature of intersections with horizontal lines. This allows determination of properties like whether or not a function is one-to-one, highlighting the utility of graphing in mathematical analysis.