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Chapter 1: Problem 70

Graph the function \(y=\sqrt{|x|}\)

Short Answer

Expert verified
The graph of \(y=\sqrt{|x|}\) is V-shaped, symmetric about the y-axis, starting at the origin and rising in both directions.

Step by step solution

01

Understand the function components

Examine the given function, \(y=\sqrt{|x|}\). This function involves two key operations: the absolute value, \(|x|\), and the square root, \(\sqrt{}\). The absolute value makes the input non-negative, and the square root then calculates the square root of this non-negative value.
02

Analyze the function's domain

Identify the domain of the function \(y=\sqrt{|x|}\). Since \(|x|\) ensures that the input to the square root is non-negative, the domain of this function is all real numbers, \(\mathbb{R}\).
03

Determine key points

Calculate key points on the graph by substituting values of \(x\). For \(x = 0\), \(|x| = 0\), so \(y = \sqrt{0} = 0\). For \(x = 1\) and \(x = -1\), \(|x| = 1\), so \(y = \sqrt{1} = 1\). For \(x = 4\) and \(x = -4\), \(|x| = 4\), so \(y = \sqrt{4} = 2\).
04

Identify symmetry

Recognize that \(y=\sqrt{|x|}\) is symmetric with respect to the y-axis because \(|x|\) treats positive and negative values of \(x\) the same way. The function values for \(x\) and \(-x\) are identical, resulting in a graph that mirrors across the y-axis.
05

Plot the graph

Using the calculated points and symmetry property, plot the graph. Start at the origin (0,0) and plot the points (1,1), (-1,1), (4,2), and (-4,2). Connect these points smoothly as the graph rises quickly near the origin and flattens out as \(x\) increases in magnitude.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
The absolute value of a number is the distance of that number from zero on the real number line. In mathematical terms, for any real number \( x \), the absolute value is denoted as \( |x| \) and is always non-negative. Here鈥檚 how it works:
  • If \( x \) is positive or zero, \( |x| = x \).
  • If \( x \) is negative, \( |x| = -x \).
The absolute value is particularly useful when dealing with expressions that require non-negative inputs. For the function \( y = \sqrt{|x|} \), the absolute value ensures that we only take the square root of non-negative numbers, since \( \sqrt{} \) is not defined for negative values in the real number system.
Square Root
The square root operation finds a number which, when multiplied by itself, gives the original number. Denoted by the symbol \( \sqrt{} \), it is only defined for non-negative numbers in the real number system. This is because multiplying a negative number by itself gives a positive result, which aligns with the property that square roots of negative numbers do not exist in the set of real numbers.
  • For \( x \geq 0 \), \( \sqrt{x} \) returns the non-negative number whose square is \( x \).
  • In our function \( y = \sqrt{|x|} \), even if \( x \) is negative, \( |x| \) is non-negative, making \( \sqrt{|x|} \) valid.
So, the square root in this context will output non-negative values, reflecting the geometric concept of half the "opening" width of an area bounded by a curve above the x-axis in its graph.
Function Symmetry
Symmetry in functions refers to the property where parts of the graph of a function mirror each other across a particular line. For the function \( y = \sqrt{|x|} \), the symmetry line is the y-axis.
In detail:
  • If \( f(x) = f(-x) \) for all relevant \( x \), the function is symmetric about the y-axis, often called an even function.
  • In \( y = \sqrt{|x|} \), the absolute value of \( x \) treats \( x \) and \(-x\) identically, since negative inputs are converted to positive before square root is applied.
This symmetry makes the graph appear like a V mirrored across the y-axis, starting at the origin and opening upwards.
Domain of Functions
The domain of a function is the complete set of possible values of the independent variable, \( x \), for which the function is defined. Delving into the domain of \( y = \sqrt{|x|} \), we note:
  • The absolute value \( |x| \) converts all \( x \) to non-negative numbers.
  • The square root function is defined for all non-negative inputs.
Thus, the domain of \( y = \sqrt{|x|} \) is all real numbers \( \mathbb{R} \), because any real number can be inputted into \( |x| \), which will allow \( \sqrt{} \) to always produce a real number output.

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