Problem 70 - 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. $$f(x)=\left\\{\begin{array}{ccc} 0 & \text { if } & x<-3 \\ -x & \text { if } & -3 \leq x<0 \\ x^{2}-1 & \text { if } & x \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

The range of the function is \(-1 \leq y < \infty\).

Step 1: Graph the first part of the function

For \(x < -3\), the function is equal to 0. Draw a horizontal line on the y-axis at \(y = 0\) that extends to the left from \(x = -3\). Make sure to use an open dot at \(x = -3\) because the function cannot equal 0 at \(x = -3\).

Step 2: Graph the second part of the function

For \(-3 \leq x < 0\), the function is equal to \(-x\). That's the equation for a line passing through the origin with a positive slope. So, plot a line segment starting from \(x = -3\) (this time with a closed dot because the function can equal -(-3) = 3 at \(x = -3\)) to \(x = 0\), making sure that the line passes through \(y = 0\) at \(x = 0\). Use an open dot at \(x = 0\) because the function cannot equal 0 at \(x = 0\)

Step 3: Graph the third part of the function

For \(x \geq 0\), the function is equal to \(x^{2}-1\). That's a quadratic function, shifted down by 1 unit. Enter this point with a closed dot, because the function can equal \((0)^2-1 = -1\) at \(x = 0\), and draw a parabola that opens upwards starting from this point.

Step 4: Determine the function's range

The range of a function is the set of all possible output values (y-values). Looking at the graph, the range of this function is \(-1 \leq y < \infty\) as the lowest point is at \(y = -1\) and the function extends upward to \(\infty\).

Key Concepts

Domain and Range

The domain of a function refers to all the possible input values, or "x" values, the function can accept. In this exercise, the domain of the piecewise function is given as \((-\infty, \infty)\). This means that the function is defined for every real number \(x\). When looking at piecewise functions, understanding the domain is crucial since the function has different equations or rules for different intervals of \(x\).On the other hand, the range of a function consists of all possible output values, or "y" values. To determine the range of a piecewise function, such as in our example, we need to carefully analyze the graph of each section of the function. In this case, the range is found to be \([-1, \infty)\). This indicates that the smallest possible value for \(f(x)\) is \(-1\), and the function can produce any value greater than or equal to \(-1\).To summarize:

• Domain: All real numbers, \((-\infty, \infty)\)
• Range: \([-1, \infty)\)

When analyzing piecewise functions, always examine each part to discover crucial information about the domain and range.

Graphing Functions

Graphing a function helps visualize how it behaves over different intervals of \(x\). In this exercise, graphing the piecewise function is essential to determine the range and understand how each segment transitions into the next.To begin with, for intervals where \(x < -3\), the function \(f(x) = 0\) is a horizontal line on the x-axis. This line extends infinitely to the left but starts at \(x = -3\) with an open dot to indicate that it's not included at that point.Next, for \(-3 \leq x < 0\), the function is linear and represented by \(f(x) = -x\). This part of the function appears as a diagonal line with a positive slope, beginning from \(x = -3\) at \(y = 3\) and continuing to \(x = 0\). It ends at this point with an open dot as the function is not defined exactly at \(x = 0\) in this segment.Finally, for \(x \geq 0\), the function becomes quadratic, expressed as \(f(x) = x^{2} - 1\). This portion is a parabola opening upwards, beginning at the point where \(x = 0\) and \(f(x) = -1\), shown with a closed dot.Graphing each segment separately is crucial:

• Horizontal Line: \(x < -3\) with \(f(x) = 0\)
• Diagonal Line: \(-3 \leq x < 0\) with \(f(x) = -x\)
• Parabola: \(x \geq 0\) with \(f(x) = x^2 - 1\)

Drawing helps you see the connection between different pieces and their combined range.

Quadratic Function

Quadratic functions are represented by the general form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the piecewise function given, the third segment \(f(x) = x^2 - 1\) is a simple quadratic function.This quadratic function exhibits some distinct characteristics:

• It opens upwards since the coefficient of \(x^2\) (which is 1 here) is positive.
• The vertex of the parabola is at the point \((0, -1)\), indicating the minimum value of the function within this interval.
• As \(x\) moves away from 0 in either direction, the value of the function increases without bound.

Because quadratic functions are symmetrical, the graph shows a nice, smooth curve that continuously rises as we move right from the vertex.An important point to note is that this segment of the piecewise function begins at \(x = 0\) due to the constraints on the piecewise structure, meaning no part of \(x^2 - 1\) will appear to the left of \(x = 0\). The significance of this quadratic section is in its contribution to the range, introducing values starting from \(-1\) to infinity. Understanding how quadratic functions are graphed can provide valuable insight into changes in a function's behavior and ensure comprehensive knowledge of piecewise functions like the one given here.