https://file.aiquickdraw.com/mathbot/file/1724763532310jsl8yspa.jpg

Let's analyze and solve problems 1 and 3 from the image.

### Problem 1
**Function:** \( y = |x| \)

The function \( y = |x| \) is known as the absolute value function. The absolute value of a number is its distance from zero on the number line, regardless of direction.

- **Graph Description:**
  - The graph of \( y = |x| \) is a V-shaped curve that opens upwards.
  - It has a vertex at the origin \((0, 0)\).
  - For \( x \geq 0 \), \( y = x \).
  - For \( x < 0 \), \( y = -x \).

- **Determining if the graph is one-to-one:**
  - A function is one-to-one if every horizontal line intersects the graph at most once.
  - The graph of \( y = |x| \) does not satisfy this criterion because horizontal lines above the x-axis will intersect the graph in two points, except at the vertex.

**Conclusion:** The function \( y = |x| \) is **not one-to-one**.

### Problem 3
**Function:** \( y = \frac{1}{x} \)

The function \( y = \frac{1}{x} \) is known as the reciprocal function. It is defined for all \( x \) except \( x = 0 \) because division by zero is undefined.

- **Graph Description:**
  - The graph consists of two branches, one in the first quadrant and one in the third quadrant.
  - As \( x \) approaches 0 from the right (positive x-axis), \( y \) approaches infinity.
  - As \( x \) approaches 0 from the left (negative x-axis), \( y \) approaches negative infinity.
  - As \( x \) increases or decreases away from zero, \( y \) approaches zero, making the x-axis a horizontal asymptote.

- **Determining if the graph is one-to-one:**
  - The graph passes the horizontal line test, as any horizontal line will intersect the graph at most once.

**Conclusion:** The function \( y = \frac{1}{x} \) is **one-to-one**.

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Would you like to explore any further details or have any additional questions about these problems?

Here are 5 related questions that could deepen your understanding:
1. What is the definition of a one-to-one function?
2. How do transformations affect the graph of the absolute value function?
3. What are the asymptotic behaviors of rational functions?
4. How would the graph of \( y = |x| + 1 \) differ from the graph of \( y = |x| \)?
5. Can you provide an example of a function that is both one-to-one and onto?

**Tip:** To determine whether a function is one-to-one, the horizontal line test is a quick and effective tool!

Explore detailed analysis of the absolute value function and reciprocal function, including their graphs, one-to-one properties, and applications. Suitable for high school students studying functions and graphing.

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