Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(x)=3 x\)

Step-by-Step Solution

Answer

The function \(f(x) = 3x\) is odd.

Step 1: Understanding Even and Odd Functions

A function is even if for every point \(x\), \(f(x) = f(-x)\). A function is odd if for every point \(x\), \(-f(x) = f(-x)\). If neither condition holds, the function is neither even nor odd.

Step 2: Evaluate \(f(-x)\)

For the given function \(f(x) = 3x\), substitute \(-x\) into the function: \[ f(-x) = 3(-x) = -3x \]

Step 3: Check for Evenness

To check if the function is even, compare \(f(x)\) with \(f(-x)\):\[ f(x) = 3x \] and \[ f(-x) = -3x \].Since \(f(x) eq f(-x)\), the function is not even.

Step 4: Check for Oddness

To check if the function is odd, verify if \(f(-x) = -f(x)\):\[ -f(x) = -3x \] and \[ f(-x) = -3x \].Since \(-f(x) = f(-x)\), the function is odd.

Step 5: Graphing the Function

Draw a graph with the \(x\)-axis and \(y\)-axis. Sketch the line \(y = 3x\), which will be a straight line passing through the origin with a slope of 3. This line exhibits symmetry about the origin, confirming it's an odd function.

Key Concepts

Graphing Linear Functions

Graphing linear functions is one of the most basic yet important concepts in algebra. A linear function is a polynomial function of degree one, usually in the form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To graph a linear function like \( f(x) = 3x \), you'll start by identifying the slope and the y-intercept. In this case, the slope \( m \) is 3, and there is no y-intercept mentioned, which means the line passes through the origin (0,0).

You can plot the graph by:

• Starting at the origin (0,0) as there's no y-intercept other than 0.
• Using the slope, which is 3, means the line rises 3 units vertically for every 1 unit it moves horizontally to the right.
• Drawing a straight line through these points, ensuring the line extends infinitely in both directions.

This results in a diagonal line that moves upwards from left to right, steeper with a greater slope.

Symmetry of Functions

The symmetry of functions refers to how a function behaves when reflected across certain lines or points.

When assessing function symmetry, you are usually looking for even, odd, or neither based on specific criteria:

• **Even Functions**: These are symmetrical about the y-axis, meaning \( f(x) = f(-x) \). An example is \( f(x) = x^2 \).
• **Odd Functions**: These are symmetrical about the origin, satisfying \( -f(x) = f(-x) \). The function \( f(x) = 3x \) is odd. This symmetry is visualized as reflection through the origin, meaning if you rotate the graph 180 degrees around the origin, it remains unchanged.
• **Neither**: If a function doesn't meet the conditions for either even or odd, it's neither.

For \( f(x) = 3x \), after comparing \( f(x) \) and \( f(-x) \), we see that it's an odd function due to this symmetry about the origin.

Function Evaluation

Function evaluation simply means substituting values into the function's equation to find the corresponding output or y-value.

Consider the function \( f(x) = 3x \). Evaluating a function involves inserting a value of \( x \) into the equation to compute \( f(x) \):

For example:

• Evaluate for \( x = 1 \): \( f(1) = 3 \times 1 = 3 \).
• Evaluate for \( x = -1 \): \( f(-1) = 3 \times (-1) = -3 \).
• Whether \( x \) is positive or negative, this simple multiplication provides the output.

Function evaluation helps in graphing, checking symmetry, and understanding the nature of the function in real-world situations. Always remember to replace \( x \) with the actual numbers when calculating \( f(x) \), and the results illustrate the line on a graph, showing how it behaves for various inputs.