Problem 10 - 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. $$ f(x)=2 x-1 $$

Step-by-Step Solution

Answer

The function \(f(x) = 2x - 1\) is not symmetrical, having no even or odd symmetry.

Step 1: Understanding the Function

The function given is a linear function, \(f(x) = 2x - 1\). This type of function will graph as a straight line. The slope of the line is 2, which means for every unit increase in \(x\), \(f(x)\) increases by 2 units. The y-intercept is -1, indicating the line crosses the y-axis at (0, -1).

Step 2: Sketching the Graph

To sketch the graph, start by plotting the y-intercept at (0, -1). From this point, use the slope of 2 to find another point on the line: go up 2 units and right 1 unit to get to the point (1, 1). Connect these points with a straight line. Extend the line in both directions to complete the graph.

Step 3: Determine Symmetry Type

A function is even if \(f(-x) = f(x)\) for all \(x\), and odd if \(f(-x) = -f(x)\) for all \(x\). Calculate \(f(-x) = 2(-x) - 1 = -2x - 1\) and \(-f(x) = -(2x - 1) = -2x + 1\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the function has no obvious symmetry.

Step 4: Confirming Symmetry Using Subsection 1.3.1

According to the criteria in Subsection 1.3.1, a linear function with a non-zero slope is generally not symmetrical. The step-by-step calculations confirm that \(f(x) = 2x - 1\) does not satisfy the conditions for even or odd symmetry, confirming from the criteria as well.

Key Concepts

Graphing Functions

Understanding how to graph functions is a big step in mathematics as it helps visualize how a function changes. Linear functions, like the one given, are among the simplest to graph. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

When graphing a linear function:

• Start by identifying the y-intercept, the point where the line crosses the y-axis. For \( f(x) = 2x - 1 \), this is \( (0, -1) \).
• Next, use the slope to find another point. The slope \( m = 2 \) tells us that for each increase in the \( x \) value by 1, the \( y \) value increases by 2.
• Plot another point using the slope, for instance, at \( (1, 1) \), and draw a straight line through these points.

By extending the line in both directions, we capture the behavior of the function over all real numbers. This visualization helps clarify aspects such as intercepts and overall direction of the line.

Even and Odd Functions

Functions can exhibit special symmetries which are described as even or odd. Determining this for a function involves using specific criteria.

For a function to be even:

• It must satisfy \( f(-x) = f(x) \) for all \( x \).
• Even functions have symmetry around the y-axis.

For a function to be odd:

• It must satisfy \( f(-x) = -f(x) \) for all \( x \).
• Odd functions have rotational symmetry around the origin, meaning the function looks the same after a 180-degree rotation.

The function \( f(x) = 2x - 1 \) is neither even nor odd as substituting \( -x \) does not satisfy either condition. This is typical for many linear functions unless the slope is zero.

Symmetry in Mathematics

Symmetry in mathematics often refers to the ways in which a function can be mirrored or rotated without changing its overall shape.

There are several types of symmetry:

• **Reflective Symmetry (Even Functions):** A function that doesn't change when mirrored across the y-axis.
• **Rotational Symmetry (Odd Functions):** A function that looks the same after a 180-degree rotation around the origin.

An additional form is **point symmetry**, where the function reflects around a point rather than a line or axis. Linear functions usually have a form that is not symmetric unless specified, making them straightforward but not typically symmetrical.

Understanding symmetry helps in predicting function behavior and simplifying complex mathematical processes. While some functions clearly show symmetry, others, like our linear function \( f(x) = 2x - 1 \), do not. Being familiar with these concepts is useful for graph analysis and function transformation.