Problem 34 - Algebra A Combined Function

Question

Graph each piecewise-defined function. $$ f(x)=\left\\{\begin{array}{rll} 4 & \text { if } & x<-3 \\ -2 & \text { if } & x \geq-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph consists of a horizontal line at 4 for \(x < -3\) and at -2 for \(x \geq -3\).

Step 1: Understanding the Problem

The function given is a piecewise function. This means that it has different expressions depending on the value of the input, \(x\). We need to determine how each piece of the function behaves and plot them accordingly.

Step 2: Identify the Parts of the Function

The function \(f(x)\) is defined by two parts: \(f(x) = 4\) if \(x < -3\) and \(f(x) = -2\) if \(x \geq -3\). These are constant functions for different intervals of \(x\).

Step 3: Plot the First Piece

For \(x < -3\), the function is constantly \(f(x) = 4\). This can be represented as a horizontal line on the graph at \(y = 4\). Since this piece is valid for \(x < -3\), draw this line starting from below \(-3\) on the x-axis and extend it to the left. End the line with an open circle at \(x = -3\) to indicate that this point is not part of this piece.

Step 4: Plot the Second Piece

For \(x \geq -3\), the function is \(f(x) = -2\). This is also a horizontal line at \(y = -2\). Draw this line starting with a closed circle at \(x = -3\) (as \(x = -3\) is included in this interval) and extend it to the right indefinitely.

Step 5: Combine and Review the Graph

Combine the two pieces on the graph to visualize the complete function behavior across the defined intervals. The function switches values from \(4\) to \(-2\) when \(x\) reaches \(-3\). Make sure the transition at \(x = -3\) is clear with an open circle for the first piece and a closed circle for the second piece.

Key Concepts

Graphing Piecewise Functions

Piecewise functions are fascinating because they exhibit different behaviors across various intervals of their domain. When we graph a piecewise function, we must represent these different behaviors clearly. This involves plotting each segment separately and ensuring continuity or discontinuity based on the conditions provided.

• Identify each piece of the function, noting the specific interval for which it is valid.
• Plot each segment as a line or curve, depending on its expression, within its given domain.
• Use open or closed circles to indicate whether endpoints are included in each segment.

In the example given, the function switches from one constant value to another at a specified point, highlighting the nature of piecewise functions.

Constant Functions

Constant functions are a special type of function where the output value remains the same irrespective of the input. They are represented as horizontal lines on a graph.

• For any input value within the function's domain, the output is always the same constant value.
• No matter how much the input changes, the output does not vary.

In the example, we have two constant functions:
1. When the input is less than \(x = -3\), the function outputs \(4\).2. For the input greater than or equal to \(x = -3\), the function gives \(-2\).
These constants create horizontal lines on the graph at \(y = 4\) and \(y = -2\), respectively.

Function Notation

Function notation is a way to symbolize the relationship between inputs and outputs in mathematical functions. It helps us understand which expression to use for a given input.

• The notation \(f(x)\) represents the function 'f' evaluated at a particular input 'x'.
• For piecewise functions, notation clarifies which expression to apply based on the input value.

In our function \(f(x) = \{\begin{array}{rl} 4, & \text{if } x < -3 \ -2, & \text{if } x \geq -3\end{array}\}\), the function notation clearly delineates that:
- Use \f(x) = 4\ if \(x < -3\), - Use \(-2\) when \(x \geq -3\).
This allows us to easily determine the output for any input within the specified domains.

Function Behavior

The behavior of a function informs us how it acts across its domain. This includes identifying changes in values, continuity, and how slope and intercepts are manifested in graph form.

• Behavior covers constancy, rise or fall, discontinuities, and intersections with axes.
• Piecewise functions demonstrate varying behavior as each piece comes with its own characteristics.

In the example of the piecewise function given:- The function is constant with distinct flat segments due to its piecewise nature.- There is a discontinuous jump from \(4\) to \(-2\) at \(x = -3\). An open circle for \(x < -3\) at \(x = -3\) shows this, switching to a closed circle for \(x \geq -3\).
This distinct switch in values provides a unique visual and analytical insight into function behaviors.