https://file.aiquickdraw.com/mathbot/file/1726065263283yajwcnos.jpg

The image contains four mathematical functions for which you are being asked to draw graphs. Here's a breakdown of each:

1. **Function 1:**
   \[
   f(x) = -x^2 + 2x - 1 \quad \forall x \in \mathbb{R}
   \]
   This is a quadratic function, which represents a parabola opening downwards since the coefficient of \(x^2\) is negative.

2. **Function 2:**
   \[
   y = |x| + x
   \]
   This function is piecewise since the absolute value will behave differently for positive and negative \(x\):
   - When \(x \geq 0\), \(y = 2x\).
   - When \(x < 0\), \(y = 0\).

3. **Function 3:**
   \[
   |x| = |y|
   \]
   This represents a pair of linear functions:
   - \(y = x\)
   - \(y = -x\)
   It forms an X-shape on the graph.

4. **Function 4:**
   \[
   y^2 = x
   \]
   This is a sideways parabola, where the square of \(y\) is equal to \(x\), meaning the graph opens to the right.

I will now prepare graphs for these functions.

Draw graphs of the following functions:
1. f(x) = -x^2 + 2x - 1 for all x in R
2. y = |x| + x
3. |x| = |y|
4. y^2 = x

This problem covers graphing various types of functions, including quadratic, absolute value, linear, and sideways parabola functions. It includes detailed breakdowns of how to approach each function and visualize their graphs. Suitable for high school students (Grades 9-12).

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