Problem 2 - Precalculus

Question

In Exercises \(1-12\), sketch the graph of the given function. State the domain of the function, identify any intercepts and test for symmetry. $$ f(x)=\frac{x-2}{3} $$

Step-by-Step Solution

Answer

The function is linear with domain all real numbers, intercepts at (2, 0) and (0, -2/3), no symmetry, and has a graph resembling a straight line.

Step 1: Determine the Domain

The domain of a function consists of all the input values (x-values) for which the function is defined. For \( f(x) = \frac{x-2}{3} \), the function is a simple linear function with no restrictions like division by zero or negatives under a square root. Therefore, the domain is all real numbers.

Step 2: Identify Intercepts

To find the y-intercept, set \( x = 0 \) and solve for \( f(x) \): \[f(0) = \frac{0-2}{3} = -\frac{2}{3}.\]So the y-intercept is \( (0,-\frac{2}{3}) \). For the x-intercept, set \( f(x) = 0 \) and solve for \( x \): \[0 = \frac{x-2}{3} \Rightarrow x - 2 = 0 \Rightarrow x = 2.\]Thus, the x-intercept is \( (2, 0) \).

Step 3: Test for Symmetry

To test for symmetry, we check for even symmetry (y-axis), odd symmetry (origin), and x-axis symmetry.- **Y-axis symmetry:** Replace \( x \) with \( -x \) and see if the function remains unchanged. \[f(-x) = \frac{-x - 2}{3} eq \frac{x-2}{3}.\]The function is not symmetric about the y-axis.- **Origin symmetry:** Check if \(-f(x) = f(-x)\). \[-f(x) = -\frac{x-2}{3} = \frac{-x+2}{3},\]which does not equal \( f(-x) = \frac{-x-2}{3} \). The function is not symmetric about the origin.The function does not exhibit any symmetry.

Step 4: Sketch the Graph

With the intercepts found in step 2, we know the line passes through \( (0, -\frac{2}{3}) \) and \( (2, 0) \). The slope of the line is \( \frac{1}{3} \), so for each step in x, the value of f(x) increases by \( \frac{1}{3} \). Draw a straight line through the intercepts with the calculated slope.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are referring to all the possible values that can be input into the function. In other words, it's the set of all permitted x-values. For the linear function \( f(x) = \frac{x-2}{3} \), determining the domain is straightforward, since it's defined for all real numbers. This is because there's no division by zero or other restrictions like square roots of negative numbers. That means you can input any real number into this function without running into any issues. Hence, the domain is all real numbers, often denoted by the symbol \( \mathbb{R} \).

Intercepts

Intercepts are the points where the graph of a function crosses the axes.

• Y-intercept: To find this point, we set \( x = 0 \) in the function and solve for \( f(x) \). Here, \( f(0) = \frac{0-2}{3} = -\frac{2}{3} \), giving us the y-intercept at the point \( (0, -\frac{2}{3}) \).
• X-intercept: This is found by setting \( f(x) = 0 \) and solving for \( x \). In this case, \( 0 = \frac{x-2}{3} \) leads to \( x = 2 \). Thus, the x-intercept is the point \( (2, 0) \).

These intercepts provide critical information about where the line crosses the axes and are essential for sketching the graph.

Graph Symmetry

Symmetry in a graph can simplify understanding of the function. Generally, we test for three types of symmetry:

• Y-axis symmetry: Replace \( x \) with \( -x \) in the function. If \( f(-x) = f(x) \), the function is symmetric about the y-axis. For \( f(x) = \frac{x-2}{3} \), \( f(-x) = \frac{-x-2}{3} \), which isn't equal to \( f(x) \).
• Origin symmetry: If \( -f(x) = f(-x) \), the function is symmetrical around the origin. For \( f(x) = \frac{x-2}{3} \), \(-f(x) = -\frac{x-2}{3} = \frac{-x+2}{3}\), which also isn't equal to \( f(-x) \).
• X-axis symmetry: This would require \( f(x) = -f(x) \), which isn’t possible for functions like this linear one.

Thus, the function \( f(x) = \frac{x-2}{3} \) doesn't exhibit any symmetry.

Slope of a Line

The slope of a line is a measure of its steepness, indicating how much \( y \) changes for a change in \( x \). For linear functions, it’s constant. In the function \( f(x) = \frac{x-2}{3} \), the slope \( m \) is \( \frac{1}{3} \). This means for every unit increase in \( x \), \( f(x) \) increases by \( \frac{1}{3} \). The rise over run is a simple way to remember slope: it’s the "rise" in \( y \) divided by the "run" along \( x \).

• A positive slope, like \( \frac{1}{3} \), means the line rises as it moves from left to right.
• If the slope was negative, the line would fall in the same direction.

Understanding the slope is crucial for graphing the line, as it indicates the direction and angle of tilt of the line.