Problem 95 - 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. $$ f(x)=\left\\{\begin{array}{l}{x^{2}-3, x<0} \\ {4 x-3, x \geq 0}\end{array} ; f(-4) ; f(0) ; f(2)\right. $$

Step-by-Step Solution

Answer

\( f(-4) = 13 \), \( f(0) = -3 \), \( f(2) = 5 \); Graph is a parabola joined with a line.

Step 1: Understanding the Function

The function \( f(x) \) is piecewise-defined with two expressions: \( f(x) = x^2 - 3 \) when \( x < 0 \) and \( f(x) = 4x - 3 \) when \( x \geq 0 \). This means one formula applies to \( x < 0 \) and another to \( x \geq 0 \).

Step 2: Evaluate the Function at \( f(-4) \)

Since \( -4 < 0 \), we use the first piece of the function: \( f(x) = x^2 - 3 \). Substitute \( x = -4 \): \( f(-4) = (-4)^2 - 3 = 16 - 3 = 13 \).

Step 3: Evaluate the Function at \( f(0) \)

Since \( 0 \geq 0 \), we use the second piece of the function: \( f(x) = 4x - 3 \). Substitute \( x = 0 \): \( f(0) = 4(0) - 3 = 0 - 3 = -3 \).

Step 4: Evaluate the Function at \( f(2) \)

Since \( 2 \geq 0 \), we use the second piece of the function: \( f(x) = 4x - 3 \). Substitute \( x = 2 \): \( f(2) = 4(2) - 3 = 8 - 3 = 5 \).

Step 5: Sketch the Graph

To sketch the graph, plot the values obtained: \( f(-4) = 13 \), \( f(0) = -3 \), and \( f(2) = 5 \). For \( x < 0 \), use \( f(x) = x^2 - 3 \), so it is a parabola opening upwards. For \( x \geq 0 \), use \( f(x) = 4x - 3 \), resulting in a straight line with a slope of 4.

Key Concepts

Function Evaluation

When dealing with piecewise functions, understanding which part of the function to use is crucial for function evaluation. A piecewise function has different expressions based on the value of the independent variable, often denoted as \( x \). In our example, the function \( f(x) \) is defined differently for \( x < 0 \) and for \( x \geq 0 \). Here are the steps to evaluate the function at specific points:

• First, determine where the value of \( x \) lies in the function's definition.
• Use the corresponding expression to compute the function's value.

For example:- At \( f(-4) \), since \(-4 < 0\), use \( f(x) = x^2 - 3 \). Calculating, you find \( f(-4) = 13 \).
- At \( f(0) \), because \( 0 \geq 0 \), employ \( f(x) = 4x - 3 \). This yields \( f(0) = -3 \).
- Lastly, at \( f(2) \), since \( 2 \geq 0 \), utilize \( f(x) = 4x - 3 \). This gives \( f(2) = 5 \).The key is recognizing which part of the piecewise function to use based on the condition that \( x \) satisfies.

Graph Sketching

Sketching graphs of piecewise functions involves plotting the different parts based on the independent variable's values. These include drawing a parabola, a line, or any other defined shape. Each segment of the graph corresponds to a piece of the function.
To sketch the graph of \( f(x) \):

• For \( x < 0 \), use the expression \( f(x) = x^2 - 3 \). This forms part of a parabola opening upwards. Evaluate points and plot them, such as \( (-4, 13) \).
• For \( x \geq 0 \), use the expression \( f(x) = 4x - 3 \). Plot points using this linear expression, such as \( (0, -3) \) and \( (2, 5) \). The line has a slope of 4, indicating it rises steeply.

Where these pieces meet at \( x = 0 \) is essential. The point \( (0, -3) \) is included in the graph, marking the transition between the parabola and the line.Accurate graph sketching requires connecting points smoothly and recognizing the function's behavior at the transition points.

Calculus Problems

Calculus challenges often involve understanding and working with piecewise functions, as they describe scenarios where a function behaves differently across its domain. Tasks may include finding limits, differentiating, or integrating piecewise functions.
Although not directly covered in the step-by-step solution, some potential calculus problems could include:

• Determining the limit as \( x \to 0 \) from either side to ensure continuity.
• Finding the derivative for each piece separately to examine where changes in slope occur and analyze any critical points.
• Calculating the integral of each part if you want to find the area under parts of the curve over a given interval, aligning with the function's definition.

By understanding how to evaluate and sketch these piecewise functions, one can tackle complex calculus problems with greater ease, ensuring a solid foundation for creating more sophisticated solutions in mathematical analysis.