Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=\frac{1}{3}(3+|x|)\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{1}{3}(3+|x|)\) is a 'V' shaped graph that intersects the y-axis at (0,1). It has two segments, the left segment defined by \(f(x)=\frac{1}{3}(3-x)\) for x<0 sloping downward and the right segment defined by \(f(x)=\frac{1}{3}(3+x)\) for x>=0 sloping upward.

Step 1: Define the Function for x<0

For x<0, the absolute value of x, |x|, can be replaced with -x (since the absolute of a negative number is its positive value). So: \(f(x)=\frac{1}{3}(3+|-x|)=\frac{1}{3}(3-x)\)

Step 2: Define the Function for x>=0

For x>=0, the absolute value of x, |x|, is simply x. Hence, the function becomes: \(f(x)=\frac{1}{3}(3+x)\)

Step 3: Plot the Function

Plot the two different equations obtained above on a coordinate plane, noting that the expression for x<0 has a negative gradient while the expression for x>=0 has a positive gradient. This will create a 'V' shape graph that is characteristic of absolute functions. Remember, the point (0,1) would always be on the graph since the function is defined at x=0.

Key Concepts

Absolute Value Functions

Absolute value functions are interesting because they express a particular kind of mathematical operation where any number is transformed into its non-negative form.
An absolute value function takes the absolute value of a number and essentially strips away any negative sign. For example, the absolute value of \(-3\) is \(3\).

The function given in the exercise, \(f(x)=\frac{1}{3}(3+|x|)\), incorporates this principle. This equation involves an absolute value operation on \(x\).

• For negative values of \(x\), such as \(-4\), the absolute value \(|x|\) becomes \(4\).
• For positive values, the absolute value is simply the number itself.

Understanding how absolute value works helps in solving and graphing this function as it influences the transformation and symmetry about the y-axis.

Piecewise-defined Functions

Piecewise-defined functions are built from multiple expressions, each valid within certain intervals of the independent variable.
With the given function, \(f(x)=\frac{1}{3}(3+|x|)\), this means breaking the function into two segments.

Here is how the function divides:

• For \(x < 0\), the function becomes \(f(x)=\frac{1}{3}(3-x)\), because the absolute value turns negative values of \(x\) into positive ones.
• For \(x \geq 0\), the function remains \(f(x)=\frac{1}{3}(3+x)\).

This split creates a piecewise function that can be graphed using different rules on different parts of the x-axis.
The process of defining such functions involves considering which formula to use according to the value of \(x\). Keeping track of these sections can be crucial for accurate graphing and analysis.

Graph Sketching

Graph sketching is a method for visually representing functions to understand their behavior across different values of \(x\).
With the function \(f(x)=\frac{1}{3}(3+|x|)\), graph sketching involves plotting points to reveal its distinctive 'V' shape.

The steps include:

• Identify key points: For this function, the key point is \( (0, 1) \) where both pieces of the piecewise function meet.
• Determine slopes: The two pieces' linear equations will have different slopes:
  • For \(x < 0\), the slope is negative since \(f(x)=\frac{1}{3}(3-x)\).
  • For \(x \geq 0\), the slope is positive as \(f(x)=\frac{1}{3}(3+x)\).
• Draw the graph: Start by plotting the key point and draw lines according to the slopes for each interval. This reflects the symmetric nature of absolute value functions, resulting in a 'V' shape.

Sketching graphs step-by-step helps reveal properties like intercepts, slopes, and continuity of the function, providing a complete picture of its behavior.