Problem 33 - Elementary and Intermediate Algebra

Question

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples \(2,3,\) and 4. $$ f(x)=|x-1| $$

Step-by-Step Solution

Answer

The graph of \(f(x)=|x-1|\) is a 'V' shape centered at (1,0) with domain \(( -\infty, \infty )\) and range \([0, \infty)\).

Step 1: Understand the Function

The function given is a piecewise linear function that uses the absolute value, making it non-linear in appearance. The expression inside the absolute value, \(x-1\), determines the horizontal shift. This function reaches its minimum value when \(x-1=0\).

Step 2: Set Up a Table of Values

Create a table to find several points for the function. Choose a range of \(x\)-values to substitute into \(f(x)=|x-1|\). Calculate the corresponding \(f(x)\) for these \(x\)-values: - When \(x=0\), \(f(x)=|0-1|=1\)- When \(x=1\), \(f(x)=|1-1|=0\)- When \(x=2\), \(f(x)=|2-1|=1\)- When \(x=-1\), \(f(x)=|-1-1|=2\)- When \(x=3\), \(f(x)=|3-1|=2\)

Step 3: Plot the Points on a Graph

Using the points from the table, plot each point \((x, f(x))\) on a coordinate graph:- Plot \((0, 1)\)- Plot \((1, 0)\)- Plot \((2, 1)\)- Plot \((-1, 2)\)- Plot \((3, 2)\)Connect the points in a 'V' shape with \((1, 0)\) as the vertex.

Step 4: Determine the Domain

The domain of the function \(f(x)=|x-1|\) includes all real numbers, because we can substitute any real number into the function: Domain: \(( -\infty, \infty )\).

Step 5: Determine the Range

The range of the function depends on the smallest possible value of \(f(x)\), which occurs at \(x=1\), where \(f(x)=0\). Since absolute values are always non-negative:Range: \([ 0, \infty )\).

Key Concepts

Piecewise Functions

A piecewise function is a mathematical expression that is defined by multiple sub-functions, each relevant over certain intervals of the domain. Absolute value functions, like \( f(x) = |x-1| \), are classic examples of piecewise functions.

This particular function is composed of two linear components joined at a point, which is where the absolute value expression inside transitions. Understanding piecewise functions is crucial because they break down complex behaviors into simpler, more manageable parts.

• When \( x-1 \geq 0 \), the function equals \( x-1 \).
• When \( x-1 < 0 \), the function equals \(-(x-1) = -x+1\).

The break in the linear behavior happens at the "critical point" where the expression inside the absolute value equals zero, i.e., \( x=1 \). Here, students can see how the absolute structure maintains non-negative outputs over their domain, showcasing a key feature of piecewise definitions.

Domain and Range

The domain and range of a function are fundamental concepts in understanding how the function behaves over its input (domain) and output (range).

For the function \( f(x) = |x-1| \), the domain encompasses all real numbers. This is because you can plug any real number into the function and get a valid output. Thus, the domain is expressed as \((-\infty, \infty)\).

The range, on the other hand, deals with possible output values. With absolute value functions, outputs are always non-negative. The function \( f(x) = |x-1| \) bottoms out at zero when \( x=1 \), and grows positively as \( x \) moves away from 1 in either direction. So, the range is \([0, \infty)\). Understanding these concepts helps you predict behavior and set up a graph correctly.

Graphing Functions

Graphing a function like \( f(x) = |x-1| \) involves visualizing its behavior through a coordinate system. This function is interesting to graph because its absolute value causes a "V" shape on the graph.

Here's how graphing unfolds:

• Plot the key points calculated from the function's value table: these include points like \((0,1)\), \((1,0)\), \((2,1)\), \((-1,2)\), and \((3,2)\).
• Connect these points smoothly to highlight the "V" shape.
• The vertex of the graph, a crucial aspect, is located at \((1,0)\). This point serves as the function's minimum value.

By laying out these details on a graph, you gain insights into both the function's symmetry and how it fulfills its domain and range conditions. Each plotted point helps piece together the broader picture of the function's structure and characteristics.