Problem 39 - Algebra A Combined Function

Question

Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. $$ f(x)=\left\\{\begin{array}{rll} x+3 & \text { if } & x<-1 \\ -2 x+4 & \text { if } & x \geq-1 \end{array}\right. $$

Step-by-Step Solution

Answer

Domain is all real numbers; range is all real numbers.

Step 1: Understand the Function

The given function is piecewise-defined, which means it has different expressions depending on the value of \( x \). We have two cases here: \( f(x) = x + 3 \) when \( x < -1 \), and \( f(x) = -2x + 4 \) when \( x \geq -1 \).

Step 2: Graph the First Piece

For \( x<-1 \), the function is defined as \( f(x) = x + 3 \). This is a linear function with a slope of 1 and a y-intercept of 3. To plot this part of the graph, find a point: for example, when \( x = -2 \), \( f(-2) = -2 + 3 = 1 \). Plot the point at \((-2, 1)\), and note that this part of the line continues leftward indefinitely but stops just before \( x = -1 \).

Step 3: Graph the Second Piece

For \( x \geq -1 \), the function is \( f(x) = -2x + 4 \). This is also a linear function, with a slope of -2 and a y-intercept of 4. Start by determining a point: when \( x = -1 \), \( f(-1) = -2(-1) + 4 = 6 \). Plot this point on the graph. This line continues rightward from \( x = -1 \), passing through points like \( (0, 4) \), and so on.

Step 4: Identify Domain

The domain of a piecewise function encompasses all possible input values (\( x \)) that the function can accept. Here, since the function is defined for all \( x \) values on the real number line with distinct expressions in different intervals, the domain is all real numbers, \( x \in (-\infty, \infty) \).

Step 5: Identify Range

The range is determined by the output values the function can produce. The line \( x + 3 \) continues downwards from \((-\infty, x < -1)\), and the line \(-2x + 4\) covers values from \( x = -1 \) onward. Thus, the range also covers all values in \( (-\infty, \infty) \), because each section of the piecewise function covers part of the y-axis completely.

Key Concepts

function graphing

When graphing piecewise functions, it's essential to understand each section separately. A piecewise function is defined by different expressions over different intervals of the domain.
To start graphing, identify the components of each piece. For example, in this exercise, the function is defined by two linear pieces:

• For values where \( x < -1 \), the function is \( f(x) = x + 3 \).
• For values where \( x \geq -1 \), the function is \( f(x) = -2x + 4 \).

Graphing involves plotting these different expressions over their specified intervals. Carefully plot points based on these equations and connect them accordingly. Make sure to consider any specified end points, ensuring that they align with the conditions of the function. Intervals involve considering open versus closed dots at junctions, which show whether a value is included in the interval or not.

domain and range

In a piecewise function, understanding the domain and range is crucial. Domain refers to all possible x-values (inputs) a function can take, while the range refers to all possible y-values (outputs) it can produce.

The domain for a piecewise function covers all intervals specified in its different parts. For the function \( f(x) \) given in the exercise, both pieces cover the entire real line between different intervals:

• The equation \( x + 3 \) is defined for \( x < -1 \).
• Meanwhile, \( -2x + 4 \) applies for \( x \geq -1 \).

Together, the domain of this piecewise function is all real numbers, \( x \in (-\infty, \infty) \).
The range, which consists of output values, is also determined by the y-values over which each piece competently spans. Since both expressions cover all values reachable along the y-axis, the range here is also all real numbers.

linear functions

Linear functions are a fundamental concept within mathematics and form the basis for many piecewise functions. They are of the form \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) the y-intercept.

In the piecewise function of this exercise, each piece is a linear function. Let's look into each one:

• For \( f(x) = x + 3 \) when \( x < -1 \), the slope \( m = 1 \) dictates it rises by one unit for every unit it moves leftward.
• For \( f(x) = -2x + 4 \) when \( x \geq -1 \), the negative slope \( m = -2 \) means it falls sharply for every step rightwards.

Each of these lines has a unique characteristic in appearance and behavior, and when graphed individually over specified intervals in a piecewise function, they illustrate how linear functions interact to form a more complex graph.

continuity in piecewise functions

Continuity in piecewise functions refers to how smooth the transition is between different sections of the function. A function is continuous at a point if there is no interruption, break, or jump at that point.

For a piecewise function to be continuous at a point where two equations meet, the value of the two pieces must coincide at that point. In the provided exercise, the point where continuity is in question is at \( x = -1 \):

• For \( x < -1 \), the function approaches \( f(-1) \) but does not include this value with the function \( f(x) = x + 3 \).
• At \( x = -1 \) and beyond, \( f(x) = -2x + 4 \) takes over and is defined at this exact point.

Due to the two sections having different endpoint definitions, the function experiences a jump at this junction. This jump indicates that the piecewise function is not continuous at \( x = -1 \) as the values do not meet seamlessly. Understanding this concept helps appreciate how piecewise functions can differ fundamentally from single equation functions in their graphing characteristics.