Problem 51 - College Algebra Essentials

Question

the domain of each piecewise function is \((-\infty, \infty)\) 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

Without seeing the actual graph, it's hard to provide the exact range. But the final answer should be the y-values of the lowest and highest points on the graph.

Step 1: Understand the Function

The function given is a piecewise function, defined differently for different intervals of \(x\). The first part is \(f(x) = 0.5x^2\) for \(x < 1\) and the second part is \(f(x) = 2x - 1\) for \(x \geq 1\). Understanding the behavior of these two types of functions on various \(x\) intervals will help in sketching the graph.

Step 2: Graph the Function

To graph the function, divide the x-axis into two sections based on the given condition \(x < 1\) and \(x \geq 1\). For \(x < 1\), use the function \(0.5x^2\), and for \(x \geq 1\), use \(2x - 1\). Also, remember to mark the point \(x = 1\) because this is where the function changes and it's important for understanding the behavior of the function. Connect the dots to draw the function curves.

Step 3: Determine the Range

The range of a function is the set of all possible output values (y-values), that we can get by substituting all possible x-values into the function. By analyzing the graph we've sketched, we should note the lowest and highest points of our graph. These points will constitute the range of the function.

Key Concepts

Graphing Functions

Understanding how to graph functions is at the core of exploring algebraic relationships visually. Imagine a function as a machine that takes an input and gives an output. When graphing a function, each input (or x-value) is paired with an output (or y-value), creating coordinates that can be plotted on a graph.

In the case of piecewise functions, like the one provided in the exercise, you graph separate functions over different intervals of the x-axis. To do this effectively:

• Identify the interval for each piece of the function.
• Graph each piece within its defined interval.
• Pay special attention to the points where the function changes, and use a closed or open dot to indicate whether the endpoint is included based on the inequality.

For the function given in the exercise, a quadratic graph is plotted for values less than 1, and a linear graph is plotted for values greater than or equal to 1. By stitching these graphs together seamlessly, one gets the complete picture of the piecewise function.

Domain and Range

The domain of a function is the complete set of possible values of the independent variable, in simpler terms, the x-values you can put into the function. For the provided function, the domain is all real numbers, denoted by \( (-\infty, \infty) \).

The range, on the other hand, is the set of all possible output values (the y-values) after we plug the domain into the function. To find the range from a graph, look for the lowest point to the highest point on the y-axis that the function will reach. If a function continues indefinitely in either direction, it may have an infinite range in that direction. For the given piecewise function, you would observe both the parabola and the line to determine the minimum and maximum y-values, keeping in mind that the graph is a combination of both sections.

Quadratic Functions

Quadratic functions are expressed in the form \( f(x) = ax^2 + bx + c \) and form a parabola when graphed. The shape of the parabola - whether it opens up or down - is determined by the sign of a. If a is positive, the parabola opens upwards; if negative, it opens downwards. Key features of the graph include:

• The vertex, which is the highest or lowest point depending on whether the parabola opens upwards or downwards.
• The axis of symmetry, a vertical line that runs through the vertex and divides the parabola into two symmetrical halves.
• The y-intercept, which is where the graph crosses the y-axis.
• The x-intercepts (if they exist), which are the points where the graph crosses the x-axis.

In the given piecewise function, the quadratic portion is \( f(x) = 0.5x^2 \) for \( x < 1 \), which is a parabola opening upwards with its vertex at the origin \( (0, 0) \) and without any x-intercepts within the defined interval.

Linear Functions

Linear functions are the simplest forms of algebraic functions and are represented by the equation of a straight line, \( y = mx + b \) where m is the slope and b is the y-intercept. The graph of a linear function is always a straight line. Some key concepts include:

• The slope is a measure of the steepness of the line; it tells us how much y changes for a change in x.
• The y-intercept is the point where the line crosses the y-axis.

For our piecewise function, the linear part is \( f(x) = 2x - 1 \) for \( x \geq 1 \). This line has a slope of 2 (rising steeply) and crosses the y-axis at -1. When graphing, starting from the point where x is 1, extend the line with the given slope to get a complete view of the linear section of the piecewise function.