Problem 67 - Precalculus Essentials

Question

The domain of each piecewise function \(i s(-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{ccc} \frac{1}{2} x^{2} & \text { if } & x<1 \\ 2 x-1 & \text { if } & x \geq 1 \end{array}\right.$$

Step-by-Step Solution

Answer

After graphing the piecewise function, observe the graph to identify the range. The range consists of all y-values from the lowest to the highest point on the graph.

Step 1: Graph the first function

Start by plotting the function \( \frac{1}{2} x^{2} \) for x values that are less than 1. Keep in mind that this shall be an upward-opening parabola, but it will only represent the part of the graph where x is less than 1.

Step 2: Graph the second function

Next, plot the function \( 2x - 1 \) for x values that are greater than or equal to 1. This is a linear function with slope 2 and y-intercept -1. It will represent the part of the graph where x is greater than or equal to 1. Make sure to include the point where x = 1, since the second function applies for x greater than or equal to 1.

Step 3: Identify the range of the function from the graph

After completing the graph of the piecewise function, observe the y-values the function takes. Find the lowest and highest points of the entire graph. The range of the piecewise function is found by considering all the y-values from the lowest to the highest point.

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial when studying piecewise functions. The **domain** of a function refers to all the possible input values (x-values). For the given piecewise function, the domain is \((-\infty, \infty)\), meaning all real numbers are included. This is typical for piecewise functions as they often cover all possible x-values by combining different segments.

The **range** describes the possible output values (y-values) you can get from the function. To find the range, you need to carefully observe the graph of the function. For each section of the piecewise function, determine the smallest and largest y-values and combine them. This way, you determine the overall range of the piecewise function.

Graphing Functions

Graphing piecewise functions involves plotting different segments for different parts of the domain. The first step is plotting each part of the function separately. Start with the first part: \(\frac{1}{2} x^{2}\), which is a parabola that opens upward for \(x < 1\).

Next, plot the linear function \(2x - 1\) for \(x \geq 1\). This segment is a straight line with a slope of 2. When graphing these parts:

• Identify any potential breaks or jumps.
• Use open or closed circles to indicate whether the endpoint is included or excluded from a part of the function.

Graphing shows how different pieces fit together in a single, continuous function.

Quadratic Functions

Quadratic functions like \(\frac{1}{2} x^{2}\) are represented by parabolas. A parabola can open upwards or downwards. In this case, it opens upwards because of the positive coefficient (\(\frac{1}{2}\)).

For piecewise functions, only a portion of the parabola is used. **Key concepts** for quadratics in piecewise functions include:

• The vertex, which is the turning point of the parabola.
• The direction it opens, determined by the sign of the coefficient.
• Understanding symmetry about the vertex line.

These principles guide how you graph and interpret the segment of the quadratic function in a piecewise definition.

Linear Functions

Linear functions are easy to recognize due to their constant rate of change. The second piece of the function, \(2x - 1\), is linear. Unlike quadratic functions, linear functions form straight lines.

**Key aspects** of linear functions include:

• The **slope**, which tells how steep the line is; here, the slope is 2.
• The **y-intercept**, which is where the line crosses the y-axis. For \(2x - 1\), this value is -1.
• Direction, determined by the sign of the slope: positive means it increases, and negative means it decreases.

When dealing with piecewise functions, ensure that you plot each segment correctly to reflect these features. This helps to visually connect the segments and understand the overall behavior of the piecewise function.