Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}x+2 & \text { if } x \leq-1 \\ x^{3} & \text { if }|x|<1 \\\ -x+3 & \text { if } x \geq 1\end{cases} $$

Step-by-Step Solution

Answer

Sketch three segments: a line \(x+2\), a cubic \(x^3\), and a line \(-x+3\).

Step 1: Understand the Piecewise Function

The function \(f(x)\) is given as a piecewise function with three conditions. 1. \(f(x) = x + 2\) when \(x \leq -1\).2. \(f(x) = x^3\) when \(-1 < x < 1\).3. \(f(x) = -x + 3\) when \(x \geq 1\).

Step 2: Sketch the First Piece

To graph \(f(x) = x + 2\) for \(x \leq -1\):- The function is a straight line with a slope of 1 and y-intercept at (0,2).- Start drawing from \(x = -1\) and include the point (-1, 1).- Extend the line to the left indefinitely.

Step 3: Sketch the Second Piece

To graph \(f(x) = x^3\) for \(-1 < x < 1\):- This is a cubic function segment.- Plot some key points: - At \(x = 0\), \(f(x) = 0\). - At \(x = -0.5\), \(f(x) = (-0.5)^3 = -0.125\). - At \(x = 0.5\), \(f(x) = (0.5)^3 = 0.125\).- Connect these points smoothly as a cubic curve. This part does not include the endpoints \(x = -1\) and \(x = 1\).

Step 4: Sketch the Third Piece

To graph \(f(x) = -x + 3\) for \(x \geq 1\):- The function is a straight line with a slope of -1 and y-intercept at (0,3).- Start drawing from \(x = 1\) and include the point (1, 2).- Extend the line to the right indefinitely.

Step 5: Combine the Pieces

Combine all the pieces on the same graph:- The segment of each piece should connect as calculated, ensuring continuity where required or noting discontinuities explicitly.- Notice that the graph is not connected at the points \(x = -1\) and \(x = 1\). - Ensure to include open or closed dots to represent non-inclusive or inclusive endpoints respectively.

Key Concepts

Graph Sketching

When it comes to graph sketching, understanding how to visualize a piecewise function is crucial. Piecewise functions are defined by different equations depending on the input value, which means you'll often have to sketch distinct parts of the function separately.

To sketch a piecewise function:

• Break down the function into its defined parts. For each part, identify the domain it covers.
• Plot each part of the function according to its specific rule, ensuring you respect the domain restrictions.
• Pay attention to the endpoints of each piece. Determine whether they are open (not included) or closed (included) based on the domain.
• Combine all segments on the same graph while noting any breaks or discontinuities.

Always remember to keep an eye on the transitions between pieces to ensure your sketch accurately represents the function's behavior.

Cubic Function

A cubic function is a polynomial of degree three and has the general form: \(y = ax^3 + bx^2 + cx + d\). In our piecewise function, one segment is defined as \(f(x) = x^3\) for \(-1 < x < 1\). This makes it a cubic function segment.

Cubic functions often have an "S" shape due to their curve, which extends upwards or downwards as the input becomes larger in magnitude. Here are some key characteristics:

• The graph passes through the origin if there's no constant term (as in this case).
• The curve tends to flatten out and change direction at certain points known as inflection points, giving it its distinctive shape.
• Cubic functions can have up to three real roots (x-intercepts) or one.
• The rate at which it increases or decreases is more pronounced than linear functions, but less intense compared to quadratic functions.

It's essential to plot a few points for accuracy and connect them smoothly to represent the cubic nature properly.

Linear Function

Linear functions are the simplest type of functions and are represented by the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

In our example, two linear functions are involved: \(f(x) = x + 2\) when \(x \leq -1\) and \(f(x) = -x + 3\) when \(x \geq 1\).

Here's what you need to know about linear functions:

• Linear functions graph as straight lines. The slope \(m\) tells you how steep the line is.
• The y-intercept \(c\) tells you where the line crosses the y-axis.
• A positive slope means the line goes upwards as it moves from left to right, while a negative slope indicates it goes downwards.
• When plotting, choose the key points and extend the line in the direction permitted by the domain.

Understanding these properties helps in accurately sketching the graph segments defined by these linear functions.

Discontinuous Functions

Discontinuities in a function occur where there are breaks or gaps in the graph. For piecewise functions, like the one we're dealing with, discontinuities often appear at the transition points between different function pieces.

In our example, the points \(x = -1\) and \(x = 1\) are points of discontinuity where the graph does not connect. Let's break this down:

• At \(x = -1\), the end of \(f(x) = x + 2\) and the start of \(f(x) = x^3\) don't connect smoothly. As such, there's a jump in the graph.
• Similarly, at \(x = 1\), the function jumps again from the cubic part to the linear part defined by \(-x + 3\).
• Discontinuities can be represented on the graph with open or closed dots at endpoints. Open dots indicate the endpoint is not included, typically signaling a jump or gap.

Understanding and identifying discontinuities are important to accurately describe the behavior of a piecewise function's graph.