Problem 39 - College Algebra

Question

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. $$ f(x)=\left\\{\begin{array}{l}{2 x-1 \text { if } x<1} \\ {1+x \text { if } x \geq 1}\end{array}\right. $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\)

Step 1: Understand the Piecewise Function

The piecewise function given is \( f(x)=\left\{\begin{array}{l}{2x-1 \text{ if } x<1} \ {1+x \text{ if } x \geq 1}\end{array}\right. \). It has two parts: \(2x-1\) applies when \(x<1\) and \(1+x\) applies when \(x \geq 1\).

Step 2: Determine Each Segment of the Function

For the portion of the function \(2x-1\), observe that this linear equation applies for \(x < 1\). The graph of \(2x-1\) is a line with a slope of 2 and y-intercept at -1. For the section \(1+x\), this linear equation covers \(x \geq 1\), describing a line with a slope of 1 and y-intercept at 1.

Step 3: Sketch the Graph

1. Begin by sketching the line \(y = 2x - 1\) for \(x < 1\). The line goes through points like (0, -1) and (1, 1), but you only sketch it up to \(x = 1\). At \(x = 1\), put an open circle at (1, 1) since this point is not included in \(y = 2x - 1\) (it's for \(x < 1\)).2. Next, sketch \(y = 1 + x\) starting at \(x = 1\) onwards. Begin with a closed dot at (1, 2) since \(x = 1\) is included in this part of the function. Extend the line rightward following its slope.

Step 4: State the Domain in Interval Notation

The function is defined for all real numbers without any breaks or gaps. Therefore, the domain of \( f(x) \) is \((-\infty, \infty)\), as there are no restrictions on \(x\) for this piecewise function.

Key Concepts

Domain in interval notation

A piecewise function can often seem complex, but defining the domain in interval notation is straightforward.
When we address the domain of a function, we identify all the possible input values of that function; essentially, any value that makes the function operate without issues.### Understanding Interval NotationInterval notation is an efficient way to communicate the set of input values. It uses brackets to depict closed intervals and parentheses for open intervals:- **Parentheses, \( () \),** indicate that an endpoint is not included.- **Brackets, \( [] \),** indicate that an endpoint is included.
For example, \((1, 5]\) means numbers greater than 1 (but not including 1) up to and including 5. ### Domain of the Given FunctionIn our piecewise function, the two segments cover different domains:- The line \( 2x - 1 \) applies when \( x < 1 \).- The line \( 1 + x \) applies when \( x \geq 1 \).
Together, these segments cover all real numbers. Since there are no breaks or exclusions, the domain in interval notation is written as \((-fty, fty)\). This means every real number is a valid input.

Graph of linear functions

Graphing linear functions helps visualize equations as straight lines on a coordinate plane. Understanding the graph of a linear function involves recognizing both slope and intercepts. ### Slope and Intercept- **Slope:** Determines how steep a line is. For a function like \( y = mx + b \), \( m \) is the slope. It tells us how much \( y \) increases when \( x \) increases by 1.- **Y-intercept:** This is the value of \( y \) when \( x = 0 \). For \( y = mx + b \), the y-intercept is \( b \).### Applying to Our FunctionFor the piecewise segments:- **Segment \(2x-1\):** This segment for \( x < 1 \) has a slope of 2 and y-intercept at -1.
It goes sharply upwards and crosses the y-axis at -1.- **Segment \(1+x\):** This depicts a more gentle slope of 1, starting at \( x = 1 \) and y-intercept at 1, emphasizing its rightward and upward direction from this point.

Constructing graphs

Constructing a graph involves plotting points and linking them with lines that represent the function's behavior across its domain.### Plotting Points- **For \(2x-1\):** Choose any \( x \) values less than 1. For instance, when \( x = 0 \), \( y = -1 \), plotting (0, -1) on the graph, guide the line towards the open point (1, 1).- **For \(1+x\):** Start at the closed point (1, 2). From here, plot more points based on chosen \( x \) values like one after 1 for clarity.### Connecting the Lines- Draw a line through each plotted point based on the function's slope. Use open or closed circles to denote whether endpoints are included or excluded.- The entire graph should reflect how the value of \( f(x) \) changes with \( x \).Through careful plotting and connecting, the piecewise nature emerges, showing different behavior segments on a continuous graph.

Interval notation

Interval notation provides a concise way to describe the range of values. It's especially handy for expressing functions' domains and ranges.### Using Interval Notation for Domains- Domains describe all valid input values for a function.- With our piecewise function, every real number is acceptable, hence \((-fty, fty)\) describes the domain. This notation efficiently communicates that no gaps exist in the allowed values of \( x \).
### General Rules of Interval Notation- Use **\(()\)** for numbers not included in the set, often due to inequality symbols like \(<\) or \(>\).- Use **\([]\)** for included numbers, associated with \(\leq\) or \(\geq\) symbols.Through mastering interval notation, one can succinctly express regulations of the functional domain and other ranges or restrictions.