Problem 41

Question

Sketch the graph and give the domain and range of the function. $$f(x)=\left\\{\begin{aligned}-1 . & x<0 \\\1 . & x>0\end{aligned}\right.$$

Step-by-Step Solution

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Answer

The graph of the function contains two horizontal lines, one at y=1 (for $x>0$), another at y=-1 (for $x<0$). The domain is all real numbers except 0, and the range of the function is $-1, 1$.

1Step 1: Understanding the Function

The function defined is a piecewise function with two pieces, each representing a constant function. In the first piece, $f(x) = -1$ for all $x < 0$. In the second piece, $f(x) = 1$ for all $x > 0$. There's no value defined for $x=0$.

2Step 2: Drawing the Graph

To sketch this, start by treating each piece as a separate entity. For $x<0$, $f(x)$ is a constant -1, which will be a straight horizontal line below the x-axis at y=-1. For $x>0$, $f(x)$ is a constant 1, which is also a straight horizontal line, but this time it is above the x-axis at y=1. There is no value at $x=0$, so we leave that as an empty point on the graph.

3Step 3: Identifying the Domain and Range

The domain of the function is all x-values that the function is defined for. Looking at the function and the graph, we see that $x$ can take any real value except 0. Therefore, the domain is $x \neq 0$. The range of the function is all the possible values for $y$ (or $f(x)$). From the graph and the function, we see that $f(x)$ only has two possible values: -1 and 1. Therefore, the range is $-1, 1$.

Key Concepts

Domain and RangeGraphing FunctionsConstant Functions

Domain and Range

Understanding the domain and range of piecewise functions is fundamental in graphing accurately and predicting how the function behaves. For the function $f(x)=\left\{\begin{aligned}-1 . & x<0 \1 . & x>0\end{aligned}\right.$, let's break it down:

Domain: This refers to all the possible input values ($x$ values) for the function. For our piecewise function, it has two separate expressions that define it for all $x$ except for $x = 0$. Therefore, the domain in set notation is $D: \{x \ | \ x \in \mathbb{R}, x eq 0\}$.
Range: This represents all the possible output values ($y$ values) that the function can produce. In our function, no matter the value of $x$ (as long as it's not zero), the outputs can only be -1 or 1. Thus, the range is $\{-1, 1\}$.

Remember, the range is dictated by the actual values the function assumes, whereas the domain is all about the values $x$ can take.

Graphing Functions

Graphing piecewise functions can seem tricky initially, but it becomes much easier when approached step-by-step. For the function $f(x)=\left\{\begin{aligned}-1 . & x<0 \1 . & x>0\end{aligned}\right.$:

Start with understanding that each piece corresponds to a simple constant function within a defined interval.
For $x < 0$, plot $f(x) = -1$. This is a horizontal line below the x-axis, where $y=-1$. Mark an open circle at $x=0$, confirming it's not included in the interval.
For $x > 0$, plot $f(x) = 1$. Here, draw a horizontal line above the x-axis, where $y=1$. Again, place an open circle at $x=0$ since $f(x)$ isn't defined at this point.

Piecewise functions often include open or closed dots to indicate whether endpoints are included or excluded. Drawing these details clarifies where each piece of the function starts and stops.

Constant Functions

Constant functions are among the simplest forms of functions but also very important. Our piecewise function here is composed entirely of constant functions.

A constant function is a function that returns the same value for all its inputs within a particular interval. For instance, $f(x)=-1$ remains $-1$ for any $x$ less than 0.
In our example, the horizontal lines in the graph precisely illustrate these constant outputs: $f(x) = -1$ and $f(x) = 1$, based on their respective domains.
Because the value of the function does not change in its interval, the graph lacks any slope—the visual representation is simply flat and parallel to the x-axis.

Constant functions help to understand more complex functions by showing regions of stability or unchanging behavior within defined boundaries.

Problem 41

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