Problem 94 - Calculus Volume 1

Question

For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. $$ f(x)=\left\\{\begin{array}{l}{4 x+3, x \leq 0} \\ {-x+1, x>0^{ ;}}\end{array} ; f(-3) ; f(0) ; f(2)\right. $$

Step-by-Step Solution

Answer

f(-3) = -9; f(0) = 3; f(2) = -1. Graph: Two lines, meeting where they are not defined at x = 0.

Step 1: Evaluate f(-3)

To find \(f(-3)\), identify which piece of the function applies. Since \(-3 \leq 0\), use \(f(x) = 4x + 3\). Substitute \(-3\) for \(x\): \(f(-3) = 4(-3) + 3 = -12 + 3 = -9\).

Step 2: Evaluate f(0)

For \(f(0)\), check which part of the function applies. \(x = 0\) falls into the case \(x \leq 0\), so use \(f(x) = 4x + 3\). Substitute \(0\) for \(x\): \(f(0) = 4(0) + 3 = 3\).

Step 3: Evaluate f(2)

For \(f(2)\), the condition is \(x > 0\), so use \(f(x) = -x + 1\). Substitute \(2\) for \(x\): \(f(2) = -2 + 1 = -1\).

Step 4: Sketch the Graph

Plot the two pieces of the function. The first piece \(f(x) = 4x + 3\) is for \(x \leq 0\). Plot the line that starts at \(f(0) = 3\) and continues leftward. The second piece \(f(x) = -x + 1\) is valid for \(x > 0\). Plot this line starting just to the right of \(x = 0\) (taking care to show a distinct point at \(x = 0\)). Connect the segments smoothly.

Key Concepts

Function Evaluation

Evaluating a piecewise function means determining the output of the function at specific values of the independent variable. In mathematics, this involves plugging in given numbers into parts of a function defined by particular conditions. For a piecewise-defined function like the one we have here, different rules apply depending on the input value.
To evaluate a function like \[\begin{cases} 4x + 3, & \text{if } x \leq 0 \ -x + 1, & \text{if } x > 0 \end{cases}\],follow these steps:

• Check which condition the given value of the independent variable meets.
• Choose the corresponding equation.
• Substitute the value into the equation.

Example: For \(f(-3)\), since \(-3\leq 0\), use \(4x + 3\) and substitute \(-3\) for \(x\), giving \(f(-3) = 4(-3) + 3 = -9\). Repeat this process for other specified \(x\) values, like \(0\) and \(2\).

Graph Sketching

The graph of a piecewise function is created by plotting segments. Each piece of the function corresponds to a specific interval on the x-axis.
For our function, the segments are:

• \(f(x) = 4x + 3\) for \(x \leq 0\): This is a line that will continue leftward from \(x = 0\).
• \(f(x) = -x + 1\) for \(x > 0\): This starts right after \(x = 0\), going rightward.

To sketch the graph:

• Start with the line \(4x + 3\) at \(x = 0\) with point \((0, 3)\) and extend it left.
• Then plot \(-x + 1\) beginning just after \(x = 0\), ensuring no overlap or confusion with previous segments.

At the juncture, the graph will not have a connecting line between \(x = 0\) for the two functions, promoting clarity in understanding piecewise transitions.

Piecewise-Defined Functions

A piecewise-defined function assigns different expressions based on the input value of the independent variable. This allows the function to capture scenarios where different conditions or rules are applicable.
Understanding such functions involves identifying:

• The conditions for each piece (e.g., \(x \leq 0\) or \(x > 0\)).
• The expressions tied to those intervals.

In practice, you'll determine which portion of the function to utilize based on the value in question. A piecewise function can efficiently model situations where a simple formula doesn't fit the entire range of contexts, like a discount applied only when a purchase exceeds a certain amount.

Independent Variable

In functions, the independent variable, often denoted by \(x\), represents the input to the function, determining which expression to employ in a piecewise-defined function. It is crucial to identify the intervals or conditions of the independent variable to correctly evaluate or sketch the function's graph.
In our example, the independent variable \(x\) comes with specific rules:

• Use \(f(x) = 4x + 3\) when \(x \leq 0\).
• Switch to \(f(x) = -x + 1\) when \(x > 0\).

The concept of the independent variable underpins not only function evaluation but also helps in understanding how graphs change their behavior across different regions of the \(x\)-axis. By accurately identifying these regions, one gains insight into the piecewise function's workings, leading to correct graph sketching and evaluations.