Problem 9 - Calculus for Biology and Medicine

Question

Sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.3.1 to check your answers. $$ f(x)=|3 x| $$

Step-by-Step Solution

Answer

The function \(f(x) = 3|x|\) is even.

Step 1: Understanding the Function

The function given is a piecewise function defined by the absolute value, i.e., \(f(x) = |3x|\). This simplifies to \(f(x) = 3|x|\) because the absolute value affects the variable \(x\). The graph of \(|x|\) is a V-shaped graph, and multiplying by 3 will vertically stretch the graph.

Step 2: Sketching the Graph

To sketch \(f(x) = 3|x|\), recognize that as \(x\) increases or decreases, \(f(x)\) increases in such a way that both 'legs' of the V-shape are straight lines symmetrical about the y-axis. This characteristic will help determine if the function is even.

Step 3: Determining Evenness or Oddness by Visual Inspection

A function is even if its graph is symmetric about the y-axis. Observing the graph of \(3|x|\), it is clear that it mirrors perfectly across the y-axis, suggesting it might be an even function.

Step 4: Checking Criteria for Even Function

To verify if \(f(x)\) is even, we need to test whether \(f(-x) = f(x)\). Substitute \(-x\) into the function: \(f(-x) = 3|-x| = 3|x| = f(x)\). This confirms that the function is even.

Key Concepts

Piecewise Function

A piecewise function is defined by multiple sub-functions, each applicable to a certain interval of the domain. The function transitions at specific points where the domain splits, offering flexibility to define a relationship that varies by segment. This makes piecewise functions particularly useful for modeling real-world situations where different rules apply over different regions.

In our case, the function is defined by the absolute value:

• For any positive or zero value of \(x\), \(f(x) = 3x\).
• For any negative value of \(x\), \(f(x) = -3x\).

The distinction **lies in the absolute value operation**, which forces all outputs to be non-negative, creating a distinct 'V' shape when graphed. The functionality of piecewise functions is clear in its capability to seamlessly define behaviors for various inputs within one comprehensive mathematical expression.

Absolute Value Graph

The absolute value function is well-known for its characteristic 'V' shape. By definition, the absolute value of any number is its distance from zero on the number line, resulting in a non-negative output. This fundamental property directly influences the resulting graph of any absolute value equation.

For function \( f(x) = |3x| \):

• The graph begins to rise sharply as \(x\) moves away from zero in either direction.
• Each 'leg' of the graph corresponds to a linear segment: \( f(x) = 3x \) **for** \( x \geq 0 \) and \( f(x) = -3x \) **for** \( x < 0 \).
• The graph is vertically stretched by a factor of 3 due to the multiplier in front of the absolute value.

This stretching means any given input results in an output three times what it would be under the basic \(|x|\) function, creating steeper slopes on the graph.

Even Function Criteria

An even function is symmetrical about the y-axis, meaning that one side of the graph mirrors the other perfectly. To ascertain if a function is even, you must check if the equation holds true that \( f(-x) = f(x) \) for all \( x \) in the function's domain.

Let's review this criterion for our function \( f(x) = 3|x| \):

• By substituting \( -x \) into the function, we find \( f(-x) = |3(-x)| = 3|x| = f(x) \).

Because this equality is verified, \( f(x) = 3|x| \) satisfies the evenness criterion. Consequently, this function produces a graph that does not change if reflected over the y-axis, confirming its even nature.

This inherent symmetry is an important property, particularly when exploring function behaviors visually using graphs.