Question

(a) Graph \(y=\sqrt{x^{2}}, y=x, y=|x|,\) and \(y=(\sqrt{x})^{2}\) noting which graphs are the same. (b) Explain why the graphs of \(y=\sqrt{x^{2}}\) and \(y=|x|\) are the same. (c) Explain why the graphs of \(y=x\) and \(y=(\sqrt{x})^{2}\) are not the same. (d) Explain why the graphs of \(y=\sqrt{x^{2}}\) and \(y=x\) are not the same.

Short answer

(a) \(y = \sqrt{x^2}\), \(y = |x|\) are the same. \(y = x\), \(y = (\sqrt{x})^2\) are the same. (b) \(\sqrt{x^2} = |x|\). (c) Different domains. (d) Absolute value function.

Step-by-step solution

- Graph the functions

Plot the graphs of the functions. - For \(y = \sqrt{x^2}\), plot the graph by noting that the square root of a square always yields the absolute value of \(x\), so the graph will resemble the \(y = |x|\) graph. - For \(y = x\), plot the graph as a straight line through the origin with a slope of 1.- For \(y = |x|\), plot the graph as a V-shaped graph that has a slope of -1 for \(x < 0\) and a slope of 1 for \(x \ge 0\). - For \(y = (\sqrt{x})^2\), note that \((\sqrt{x})^2\) simplifies to \(x\), so the graph will be the same as \(y = x\).

- Compare and note identical graphs

Notice that the graphs of \(y = \sqrt{x^2}\) and \(y = |x|\) are identical. This is because \(\sqrt{x^2} = |x|\). Also, observe that the graphs of \(y = x\) and \(y = (\sqrt{x})^2\) are identical since \((\sqrt{x})^2 = x\) for all \(x\). So the identical graphs are: - \(y = \sqrt{x^2}\) is the same as \(y = |x|\).- \(y = x\) is the same as \(y = (\sqrt{x})^2\).

- Explain why \(y = \sqrt{x^2}\) and \(y = |x|\) are the same

The graph of \(y = \sqrt{x^2}\) is the same as the graph of \(y = |x|\) because \(\sqrt{x^2}\) is defined as the absolute value of \(x\). This means \(\sqrt{x^2} = |x|\) for all real values of \(x\).

- Explain why \(y = x\) and \(y = (\sqrt{x})^2\) are not the same

While \((\sqrt{x})^2\) simplifies to \(x\), the domain of \(\sqrt{x}\) is restricted to non-negative values of \(x\) (i.e., where \(x \ge 0\)). Therefore, the graph of \(y = (\sqrt{x})^2\) only covers the right half of the graph of \(y = x\), which spans both positive and negative values of \(x\).

- Explain why \(y = \sqrt{x^2}\) and \(y = x\) are not the same

The graph of \(y = \sqrt{x^2}\) is not the same as \(y = x\) because \(\sqrt{x^2}\) always produces non-negative values, effectively making it equal to the absolute value of \(x\). Therefore, while \(y = x\) includes negative values for \(x < 0\), \(y = \sqrt{x^2}\) only includes their positive counterparts, mirroring \(y = |x|\).

Key concepts

Understanding Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. Hence, it’s always non-negative. Mathematically, the absolute value of a number x is denoted as |x|. For example,

• |3| = 3
• |-3| = 3

This property is crucial in graphing because it defines the shape of certain functions. Consider the function y = |x|.
The graph of this function is V-shaped with a vertex at the origin (0,0) and symmetric along the y-axis. It forms two linear segments:

• The segment y = x for x ≥ 0.
• The segment y = -x for x < 0.

The equation y = √(x^2) essentially produces the same result as y = |x| since the square root of x squared gives the absolute value of x. Thus, these two graphs are identical.

Exploring Domain Restrictions

Domain restrictions limit the set of input values (x-values) for which a function is defined. Understanding domain restrictions is essential when comparing functions.
Consider the function y = √x. The square root function is only defined for non-negative values of x, so its domain is x ≥ 0.
When squaring this, we get y = (√x)^2 = x.
While this simplifies to y = x, keep in mind the domain restriction: x must be non-negative.

• The graph of y = x spans both positive and negative values of x.
• In contrast, the graph of y = (√x)^2 only covers x ≥ 0.

Therefore, even though these functions look the same algebraically, their graphs differ because of domain restrictions.
This is why understanding domain restrictions is vital for accurately interpreting graphs and functions.

Comparing Functions

Comparing functions often reveals key differences and similarities in their behavior and graphs. To determine if functions are equivalent, you can investigate both their algebraic forms and their graphical representations.
Let's consider these observations:

• y = √(x^2)
• y = |x|
• y = x
• y = (√x)^2

The graphs of y = √(x^2) and y = |x| are identical because √(x^2) equals |x| for all x-values. Both functions ignore the sign of x and only give non-negative results. However, y = x includes both positive and negative x-values, making its graph a straight line through the origin. On the other hand, y = (√x)^2 simplifies to y = x, but the restriction on the domain of the square root function means it only covers non-negative x-values.
Therefore, although the algebraic forms might seem similar, the domain restrictions and the behavior of the functions distinguish their graphs.

Square Root Functions

The square root function is a fundamental concept in mathematics. The function y = √x gives the principal (non-negative) square root of x. It's important to remember the domain of this function is limited to x ≥ 0 because you cannot take the square root of a negative number in the set of real numbers.
Consequently, when graphing y = √x, the plot starts from the origin (0,0) and extends to the right, creating a curve that increases gradually.

• For example,
  • √0 = 0
  • √1 = 1
  • √4 = 2
  • √9 = 3

The function y = (√x)^2, which seems to simplify to y = x, retains the restriction from its square root component, meaning it’s only defined for x ≥ 0.
In contrast, y = x does not have this restriction, allowing negative x-values, resulting in different graphical representations. Remember this distinction when dealing with square root functions and their transformations to avoid any confusion.