Problem 70 - 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 \( f(x) = x \cdot |x| \) is not one-to-one.

Step 1: Understanding the function

The function given is \( f(x) = x \cdot |x| \). To understand this function, we need to consider the absolute value. When \( x \) is positive or zero, \(|x| = x\), thus \( f(x) = x^2 \). When \( x \) is negative, \(|x| = -x\), thus \( f(x) = -x^2 \).

Step 2: Analyzing the expression for different intervals

We can now split the function into different cases:1. **If** \( x \geq 0 \), then \( f(x) = x^2 \).2. **If** \( x < 0 \), then \( f(x) = -x^2 \).This gives two different expressions of \( f(x) \) depending on the sign of \( x \).

Step 3: Sketching the graph of the function

To sketch the function, plot \( f(x) = x^2 \) for \( x \geq 0 \) (a parabola opening upwards), and \( f(x) = -x^2 \) for \( x < 0 \) (a parabola opening downwards). The graph will look like two parabolic curves joined at the origin (0,0), one upward opening from the right and one downward opening from the left.

Step 4: Determining if the function is one-to-one

A function is one-to-one if every horizontal line intersects the graph at most once. Consider horizontal lines intersecting both upward and downward parts of the graph; this will happen for any horizontal line with \( y eq 0 \). Therefore, the function is not one-to-one, because these intersections prove it is not unique.

Key Concepts

Absolute Value

The concept of absolute value is vital when understanding the behavior of a function like \( f(x) = x \cdot |x| \). Absolute value, denoted \(|x|\), measures the distance of a number from zero on the number line:

• If \( x \geq 0 \), \(|x| = x\).
• If \( x < 0 \), \(|x| = -x\).

This means that \(|x|\) essentially strips away the sign of \( x \), portraying both positive and negative numbers as positive. In our function, \( f(x) = x \cdot |x| \), the absolute value impacts the function based on the sign of \( x \).

• When \( x \geq 0 \), \( f(x) = x^2 \).
• When \( x < 0 \), \( f(x) = -x^2 \).

This kind of split based on the sign of \( x \) is typical when absolute value is involved, allowing us to effectively analyze different sections of the function's behavior.

Parabola

Parabolas are U-shaped graphs that are fundamental in algebra and calculus. The graph of a parabola can either open upwards or downwards. This behavior is clearly seen in the function \( f(x) = x \cdot |x| \):

• For \( x \geq 0 \), the function simplifies to \( f(x) = x^2 \). This creates a parabola that opens upwards, originating from the origin \((0,0)\).
• For \( x < 0 \), the function is \( f(x) = -x^2 \). This results in a parabola opening downwards, also originating from the origin.

Understanding which way the parabola opens is key to determining how the graph behaves across the x-axis. With the given function, it is important to recognize that the point of intersection between the two sections of parabolas is the origin, which serves as a pivot point between their directions.

Graph Sketching

Sketching graphs can help you visualize functions more effectively, especially when determining properties like being one-to-one. For our function \( f(x) = x \cdot |x| \), sketching involves drawing two different curves connected at the origin. Here’s how it can be done:

First, plot the curve \( f(x) = x^2 \) for \( x \geq 0 \). This part of the graph shows a parabola opening upwards from the origin, illustrating how the function increases as \( x \) becomes larger.
Then, sketch the curve \( f(x) = -x^2 \) for \( x < 0 \). This portion of the graph opens downwards and decreases as \( x \) becomes more negative.

It's essential to note how these two parabolic curves meet at the origin, forming a continuous graph. However, this meeting also highlights why the function is not one-to-one; horizontal lines intersect both the upward and downward curves at different sections. This kind of insight is why graph sketching proves useful in analyzing functions thoroughly. Using visualizations often clarifies more complex details that algebraic expressions alone might not immediately reveal.