Problem 66 - 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. $$f(x)=\left\\{\begin{array}{rll} 4 & \text { if } & x \leq-1 \\ -4 & \text { if } & x>-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the function is two horizontal lines at \(y = 4\) and \(y = -4\). The range of the function is \(-4\) and \(4\).

Step 1: Understand the Piecewise Function

Firstly, notice that the given function is a piecewise function. It is composed of two 'pieces'. One with an output of \(4\) when \(x \leq -1\) and the other with an output of \(-4\) when \(x > -1\). These separate pieces make up the whole function.

Step 2: Graph the Function

Plot the two pieces on the graph separately. For \(x \leq -1\), all the y values are \(4\). To graph this, draw a horizontal line at \(y = 4\) from \(x = -1\) and towards the negative x direction. Since \(x\) is included in \(-1\), fill in the dot on the line at \(x = -1\). For \(x > -1\), all the y values are \(-4\). To graph this piece, draw a horizontal line at \(y = -4\) from right of \(x = -1\). With \(x>-1\), the point at \(x = -1\) is not included in the line, so put an open circle at \(x = -1\).

Step 3: Determine the Range

The range of a function is the set of all possible output values (y-values) in the function. From the graph, it can be seen that there are only two possible y-values for all x, those are \(4\) and \(-4\). Thus, the range of the given function is \(-4\) and \(4\).

Key Concepts

Understanding the Domain

The domain of a piecewise function includes all the possible input values (x-values) that the function can accept. For the given function, the domain is from negative infinity to positive infinity, written as \((-\infty, \infty)\). This means the function is defined for every real number. Here's why:

• Piecewise functions can have different rules for different intervals.
• In this case, the function has two distinct rules:
• When \(x \leq -1\), the function value is \(f(x) = 4\).
• When \(x > -1\), the function value is \(f(x) = -4\).

Understanding the domain helps us know where the function is applicable and lets us graph it correctly.

Understanding the Range

The range of a function refers to all the possible output values (y-values) it can produce. For the piecewise function given, the range is determined by looking at the y-values achieved within the domain.

• Two different y-values are produced:
• When \(x \leq -1\), \(f(x) = 4\).
• When \(x > -1\), \(f(x) = -4\).
• This implies the range is \(-4, 4\).

Visualizing the graph reveals how the range is created by different sections of the function, and knowing the range is vital for understanding what outputs are achievable.

Graphing Piecewise Functions

Graphing piecewise functions might seem challenging at first, but it's about understanding each piece separately and then combining them on the graph.

Here's a simple approach:

• For \(x \leq -1\), draw a horizontal line at \(y = 4\).
• Include a solid dot at \(x = -1\) to show the point is part of the function.
• For \(x > -1\), draw a horizontal line at \(y = -4\).
• Use an open circle at \(x = -1\) because this point isn't included in the second piece.

Graphing each piece separately and understanding where they overlap helps to clarify how the function behaves, providing a complete picture of the piecewise function.