Problem 54 - 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. $$ f(x)=\left\\{\begin{array}{ccc} 0 & \text { if } & x<-3 \\ -x & \text { if } & -3 \leq x<0 \\ x^{2}-1 & \text { if } & x \geq 0 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph of the function consists of three parts. For \(x < -3\), it's a horizontal line at \(y = 0\). For \(-3 \leq x < 0\), it's a straight line with a negative slope of 1, and for \(x \geq 0\), it's a quadratic function shifted down by 1 unit. The range of the function is \([-1, \infty)\).

Step 1: Graph the first piece of the function

The first piece of the function is \(f(x) = 0\) for \(x < -3\). The graph of this piece is a horizontal line along the x-axis from \(-\infty\) to -3, not including -3.

Step 2: Graph the second piece of the function

The second piece of the function is \(f(x) = -x\) for \(-3 \leq x < 0\). The graph of this piece is a straight line with a negative slope of 1. It starts at the point (-3, 3) and ends at the point (0, 0). It includes the point (-3, 3), but does not include the point (0, 0).

Step 3: Graph the third piece of the function

The third piece of the function is \(f(x) = x^2 - 1\) for \(x \geq 0\). This is the graph of a quadratic function shifted down by 1 unit. The graph starts at the point (0, -1) and continues upwards as x increases.

Step 4: Combine the graphs

Combine the three pieces to create the graph of the piecewise function. Remember to include all of the points and their correct boundaries.

Step 5: Determine the range

Based on the graph produced, the range of the function can be obtained. The function starts at -1 as the lowest point and moves upwards as x increases with no limit. Thus the range is \([-1, \infty)\).

Key Concepts

Graphing Functions

Graphing functions, especially piecewise functions, require breaking down the function into its separate parts. Each part is defined over a specific interval of the domain. Here’s how you should approach graphing a piecewise function:

• Identify each piece of the function and its respective interval by looking at the function definition.
• Graph each individual piece carefully, respecting whether endpoints are included or not.
• Use open or closed dots to indicate whether a value is included (closed dot) or not (open dot).

In the given example, you have three pieces of the function: a constant piece, a linear piece, and a quadratic piece. The graphing process involves connecting these pieces, such that they respect their given intervals and conditions. Remember to label key points on your graph for clarity.

Function Domain

The domain of a function is the set of all possible input values (often denoted as 'x') that the function can accept. In the case of piecewise functions, domains can sometimes be restricted to specific intervals for each piece.For example, in the given function:

• The first piece, defined by the condition \(x < -3\), has the domain \((-\infty, -3)\).
• The second piece, defined by \(-3 \leq x < 0\), covers the domain \([-3, 0)\).
• The third piece, for \(x \geq 0\), spans over \([0, \infty)\).

However, when combined as a piecewise function, the entire domain is the union of all individual domains, giving us the full range of possible inputs from \(-\infty\) to \(\infty\).Clearly defining domains helps in understanding which piece of the function to use for any given input.

Function Range

The range of a function is the set of all possible outputs (often denoted 'y') that the function can produce. For piecewise functions, albeit the domain may be unrestricted, the range depends on the outputs of each interval.To determine the range:

• Look at the outputs for each interval defined by the function.
• Evaluate from the lowest possible value of y to the highest based on your graph.

In this particular exercise, we see from the graph that the function's outputs include:

• Constant zero values leading up to, but not including, \(-3\).
• Values descending from 3 to 0 for the linear piece.
• For values beyond \(x = 0\), the quadratic portion takes over, and the values increase infinitely after starting at \(-1\).

Thus, the overall range is \([-1, \infty)\), indicating the lowest value is \(-1\) and continues upwards without bound.

Quadratic Functions

Quadratic functions are polynomial functions of the form \(f(x) = ax^2 + bx + c\). They're characterized by a parabolic graph that opens either upwards or downwards.In the piecewise function given, the quadratic part is \(f(x) = x^2 - 1\), starting at \(x = 0\). Key features of quadratic functions include:

• The vertex, which is the point where the parabola changes direction. For \(x^2 - 1\), the vertex at \((0, -1)\) indicates the lowest point because the parabola opens upwards.
• A symmetrical shape about the vertex line.
• A range that can either be from a minimum value to infinity or from negative infinity to a maximum value, depending on the direction the parabola opens.

Understanding the properties of quadratic functions helps greatly in graphing and analyzing the behavior of such functions in piecewise forms. In this scenario, the quadratic section begins the range at \(-1\) and rises continually, contributing significantly to the function’s range.