Problem 60 - Algebra and Trigonometry

Question

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=x \cdot|x|\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = x \cdot |x| \). This means that for positive \( x \), \( f(x) = x^2 \) and for negative \( x \), \( f(x) = -x^2 \). Essentially, this function is piecewise, with different definitions depending on the sign of \( x \).

Step 2: Analyze and Plot Points

Determine a few key points to help graph the function. For \( x > 0 \), use points such as \( (1, 1), (2, 4), (3, 9) \) and for \( x < 0 \), use points such as \( (-1, -1), (-2, -4), (-3, -9) \). Each side is a parabola, upward for positive \( x \) and downward for negative \( x \).

Step 3: Draw the Graph

Sketch the graph based on the points plotted. The graph is a parabola with a vertex at the origin. For \( x > 0 \), the parabola opens upwards, and for \( x < 0 \), the parabola opens downwards, forming a 'v' shape with the point at the origin.

Step 4: Evaluate One-to-One Property

A function is one-to-one if every horizontal line intersects the graph at most once. Observe the graph: horizontal lines will intersect twice, once in the positive \( x \) region and once in the negative \( x \) region (except at the origin). This tells us that \( f(x) = x \cdot |x| \) is not a one-to-one function.

Key Concepts

Piecewise Functions

Piecewise functions are functions that have different expressions depending on the value of the input variable, usually denoted as "x."
They behave differently in different intervals of their domains.

• For example, our given function is \(f(x) = x \cdot |x|\), which behaves differently for positive and negative values of \(x\).
• For \(x > 0\), \(f(x) = x^2\); this translates to the positive side of the x-axis.
• Conversely, for \(x < 0\), \(f(x) = -x^2\); this reflects the function over the y-axis for negative values.

This reflective quality helps us analyze such functions better, as we can compartmentalize the behavior of the function in its respective domain.
Understanding how piecewise functions work not only aids in graphing but also in determining properties such as continuity and differentiability.

One-to-One Functions

A one-to-one function is a type of function where each element of the range corresponds to exactly one element of the domain. This is also known as being "injective."
For a function to be one-to-one, no horizontal line should intersect the graph of the function more than once.

• In the case of \(f(x) = x \cdot |x|\), the graph shows multiple intersections with horizontal lines, indicating it's not one-to-one.
• The test involves checking the values: if \(f(a) = f(b)\) implies \(a = b\), then the function is one-to-one.

So for any pair of distinct inputs, their outputs must also be distinct for the function to qualify. The lack of this property in our piecewise function tells us it's not injective.
Thus, it's important to visualize and evaluate one-to-one properties especially when dealing with piecewise functions.

Parabolas

Parabolas are smooth, curve-shaped symmetric graphs that look like an open bowl.
The general form is \(y = ax^2 + bx + c\), where the coefficient \(a\) determines the direction of the opening.

• In our piecewise function, segments of the graph for positive and negative values of \(x\) behave as parabolas.
• The parabola for \(x > 0\) is \(y = x^2\), opening upwards.
• For \(x < 0\), \(y = -x^2\), the parabola opens downwards.

This gives us two mirror-image parabolas meeting at the vertex at the origin.
The vertex point is crucial as it gives the minimum or maximum depending on the direction the parabola opens. Understanding the properties of parabolas further aids in graphing and analysis.

Function Graphing

Function graphing involves plotting the relationship between the input variable "x" and the corresponding output "f(x)" on a coordinate system.
This visual representation helps in understanding the function's behavior.

• While graphing \(f(x) = x \cdot |x|\), we can plot key points for each segment of the piecewise definition.
• Key points like \((1, 1), (2, 4), (3, 9)\) for \(x > 0\) and \((-1, -1), (-2, -4), (-3, -9)\) for \(x < 0\) highlight its parabolic nature.

The graph results in a 'v'-shaped curve centered at the origin, showcasing how the parabolas mirror each other.
Graphing aids in visualizing the entire domain and range of the function, illustrating properties like symmetry and intercepts.