Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ f(x)=|2 x| $$

Short answer

The function is even and its graph is a V-shaped graph symmetric about the y-axis.

Step-by-step solution

Define Even and Odd Functions

An even function is symmetric about the y-axis, meaning that \( f(x) = f(-x) \) for all \( x \) in the domain. An odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain. If a function doesn't follow these rules, it is neither even nor odd.

Apply Even-ness Condition

For the function \( f(x) = |2x| \), we need to check if \( f(-x) = f(x) \). Substitute \( -x \) into the function: \( f(-x) = |2(-x)| = |2x| = f(x) \). The function satisfies the condition for being even.

Check Odd Function Condition

To confirm it's not odd, we ensure that \( f(-x) eq -f(x) \). We already found \( f(-x) = |2x| \), and \( -f(x) = -|2x| \). Clearly, \( |2x| eq -|2x| \) for all \( x \), confirming it is not odd.

Conclude Even or Odd

The function \( f(x) = |2x| \) is even because it satisfies the condition \( f(x) = f(-x) \).

Graph the Function

The graph of \( f(x) = |2x| \) is a V-shaped graph symmetric about the y-axis. It passes through the origin \((0,0)\) and opens upwards with a slope of 2 on either side of the y-axis.

Key concepts

Function Symmetry

Function symmetry is a crucial idea in math, involving how a function can reflect across its axes. There are two main types of symmetry: even and odd.

When we talk about **even functions**, we are focusing on symmetry around the y-axis. Imagine a mirror placed at the y-axis: if both sides of the function reflect perfectly across this mirror, then it's even. Essentially, for any x-value, substituting \(x\) and \(-x\)\ into the function will yield the same result: \(f(x) = f(-x)\).

On the other hand, **odd functions** are symmetric about the origin. This means they have a sort of rotational symmetry. For a function to be odd, rotating the graph 180 degrees around the origin results in the same graph. The defining property here is \(f(-x) = -f(x)\).

If a function doesn't match either condition, then it's neither even nor odd. This makes symmetry a great tool to simplify our understanding of functions! For the function from the exercise, \(f(x) = |2x|\), we've determined it's even since \(f(x) = f(-x)\).

Absolute Value Function

The **absolute value function** takes whatever number inside and turns it into its non-negative counterpart. It's like measuring the distance from zero on the number line, always resulting in a positive value or zero.

When we look at \(f(x) = |2x|\), we're essentially saying, "Take \2x\ and strip away any negative sign." This is why absolute value functions are often associated with v-shaped graphs. At the core, they will always be either zero or positive.

This feature leads to straightforward checking for symmetry as absolute value functions are often even. That's because flipping an input from positive to negative doesn't change the output—it remains positive! If we think about it graphically, the function \(f(x) = |2x|\) is bound tightly, being even, thanks to this consistent positive-valued output it generates around zero.

Graph Sketching

**Graph sketching** is like crafting a visual story of a function and can tell us so much about its behavior with just a glimpse.

For \(f(x) = |2x|\), picture the base graph \(|x|\) but stretched horizontally by a factor of 2. The result is a sharp, v-shaped graph, symmetrical around the y-axis. This symmetry is what makes graph sketching particularly neat for even functions.

During sketching:

• The vertex of the v-graph is at the origin, point \( (0,0) \).
• From the origin, as x increases or decreases, the distance between \(y\) and \(x\) is proportional—here, it's \2x\. This gives the slope of 2 on both sides.
• Because the graph is v-shaped, it clearly opens upwards, creating that crisp symmetry on either side of the y-axis.

Graph sketching like this can provide intuition at a single glance—it outlines how the function behaves and aids in understanding other properties, like symmetry and value transformations.