Problem 29 - Algebra A Combined Function

Question

Graph each piecewise-defined function. $$ f(x)=\left\\{\begin{array}{lll} 4 x+5 & \text { if } & x \leq 0 \\ \frac{1}{4} x+2 & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph consists of two lines that meet at \((0,5)\) for \(x\leq0\) and start from \((0,2)\) for \(x>0\).

Step 1: Identify the function pieces

The given function is a piecewise function with two parts. For \( x \leq 0 \), the function is \( 4x + 5 \). For \( x > 0 \), the function is \( \frac{1}{4}x + 2 \). Each part covers a different domain of the input variable \( x \).

Step 2: Graph the first piece \( 4x + 5 \) for \( x \leq 0 \)

Start by identifying a few key points for the line \( 4x + 5 \). At \( x = 0 \), \( f(x) = 5 \). At \( x = -1 \), \( f(-1) = 4(-1) + 5 = 1 \). Plot these points and draw a solid line for all \( x \leq 0 \). Use a solid dot at \( (0,5) \) since \( x = 0 \) is included.

Step 3: Graph the second piece \( \frac{1}{4}x + 2 \) for \( x > 0 \)

Identify a few key points for the line \( \frac{1}{4}x + 2 \). At \( x = 1 \), \( f(1) = \frac{1}{4}(1) + 2 = 2.25 \). At \( x = 2 \), \( f(2) = \frac{1}{4}(2) + 2 = 2.5 \). Plot these points and draw a dashed line for all \( x > 0 \). Use an open circle at \( (0,2) \) because \( x = 0 \) is not included for this piece.

Step 4: Combine the graphs

Overlay the two graphs onto the same coordinate plane. The first part of the function covers the left side (\( x \leq 0 \)), and the second part covers the right side (\( x > 0 \)). Ensure transitions at \( x = 0 \) are consistent with part-specific domain constraints (solid for first part, open for second).

Key Concepts

Graphing Functions

Graphing functions is like drawing a picture of a function on a piece of graph paper. You plot points calculated from the function's formula and connect them to see the shape they make. In piecewise functions, like the one given in our exercise, you have different formulas for different parts of the domain. Each formula gives you a part of the complete graph.

When graphing piecewise functions, it is important to pay attention to the domains of each piece. For example, in the solution provided, the piece for \(4x + 5\) is graphed using a solid line for all \(x \leq 0\). The line continues indefinitely to the left. On the other hand, the second piece \(\frac{1}{4}x + 2\) is graphed as a dashed line starting right at \(x > 0\), because this part only applies to positive \(x\) values.

Always remember:

• Plot points using your equations.
• Draw solid lines where the function is continuously defined.
• Use dashed lines or open circles where the function doesn't include boundary points.

Mastering piecewise graphs is a great step in understanding more complex mathematical concepts.

Function Domains

In mathematics, a domain is the set of all possible input values (often \(x\)-values) for which a function is defined. Domains help dictate which parts of a graph you can draw using a particular function rule. When working with piecewise functions, each piece has its own distinct domain.

In our exercise, there are two pieces with different domains:

• The domain for \(4x + 5\) is \(x \leq 0\).
• The domain for \(\frac{1}{4}x + 2\) is \(x > 0\).

Always check the conditions or limits of \(x\) provided in a piecewise function. They tell you exactly where one piece stops and the other starts.

Understanding domains is crucial because it helps you know where to draw each portion of the graph and what kinds of dots or lines to use at domain boundaries (solid or open circles). This ensures you accurately portray the function's behavior within the specified intervals.

Linear Functions

Linear functions are functions that create a straight line on a graph. They are expressed in the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) tells you how steep the line is, and the y-intercept \(b\) tells you where the line crosses the y-axis.

Let's look at each piece of the piecewise function from our exercise:

• The first piece \(4x + 5\) has a slope of 4 and a y-intercept of 5. This tells you that for every step you move right on the \(x\)-axis, the function value increases by 4.
• The second piece \(\frac{1}{4}x + 2\) has a much smaller slope of \(\frac{1}{4}\). It increases much more slowly, and starts crossing the y-axis at 2.

With this understanding, you can predict how changes in the equations affect their graphs. Recognizing linear patterns is not only useful for solving equations but also for understanding how data behaves in real life scenarios. Linear functions are fundamental building blocks in mathematics.