Problem 7 - Calculus for Biology and Medicine

Question

In Problems \(7-12\), 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 1.2.1 to check your answers. $$ f(x)=2 x $$

Step-by-Step Solution

Answer

The function \(f(x) = 2x\) is odd.

Step 1: Understanding Function Symmetry

First, let's recall what it means for a function to be even, odd, or neither. A function is even if for every point \(x\), \(f(x) = f(-x)\). This results in a graph symmetric about the y-axis. A function is odd if for every point \(x\), \(f(-x) = -f(x)\). This creates symmetry about the origin. For neither symmetry, neither condition holds.

Step 2: Testing for Even Function

For the function \(f(x)=2x\) to be even, it must satisfy \(f(x) = f(-x)\). Let's evaluate this.\[\begin{align*}f(x) &= 2x,\f(-x) &= 2(-x) = -2x.\end{align*}\]Clearly, \(f(x) eq f(-x)\), so it is not an even function.

Step 3: Testing for Odd Function

Now determine if \(f(x)=2x\) is odd by checking if \(f(-x)=-f(x)\).\[\begin{align*}f(x) &= 2x,\f(-x) &= -2x.\end{align*}\]Here, \(f(-x) = -f(x)\), so the function is odd.

Step 4: Confirm Symmetry Using Graph

Let's verify this visually by sketching the graph of \(f(x) = 2x\). The graph is a straight line passing through the origin with a positive slope, indicating symmetry about the origin. This confirms our algebraic test that the function is indeed odd.

Step 5: Conclusion

Based on both the algebraic test and the graphical representation, we conclude that the function \(f(x) = 2x\) is odd, as it exhibits symmetry about the origin.

Key Concepts

Even Function

An **even function** is characterized by its symmetrical property about the y-axis. This means for every point \(x\) on the graph, you will find another point at \(-x\) with the same \(y\)-value. Simply put, if you were to fold the graph along the y-axis, both halves would match perfectly. This is because the defining equation for an even function is \(f(x) = f(-x)\).
A classic example of an even function is \(f(x) = x^2\), where for every positive \(x\), the negative counterpart has the same output. This symmetry makes it easy to identify when sketching the graph.

• Identifying even functions involves checking the equality \(f(x) = f(-x)\).
• Graphically check if the graph nicely reflects over the y-axis.

Odd Function

If a function is **odd**, it features a different kind of symmetry known as origin symmetry. This means that flipping the graph upside down over the origin will look the same as its original graph. The algebraic criterion for an odd function is \(f(-x) = -f(x)\).
A good example of an odd function is \(f(x) = x^3\). Here, positive and negative inputs result in outputs that are negatives of each other. The function \(f(x) = 2x\) from our exercise is also odd, as we can see when plotting it: \(f(-x) = -2x\) matches \(-f(x)\).

• The defining characteristic is that the graph appears the same when rotated 180 degrees around the origin.
• Practically check by ensuring \(f(-x) = -f(x)\) holds true for your function.

Graph Sketching

**Graph sketching** is a valuable skill that helps visualize functions and determine properties such as symmetry. When graphing a linear function like \(f(x) = 2x\), you start with a basic understanding of its slope and intercept.
Here's a step-by-step on how to do it:

• Identify the slope: In \(f(x) = 2x\), the slope \(m = 2\) indicates how steeply the line rises with a positive value.
• Determine intercepts: Since there's no constant term, the y-intercept is at the origin \((0,0)\).
• Plot additional points: For better accuracy, calculate additional points using different \(x\)-values.

Once plotted, connect the points; in this case, you get a straight line through the origin. Observing this line helps confirm the function's properties, showing, for instance, the origin symmetry revealing its odd nature.