Problem 63 Graph the indicated functions. ... [FREE SOLUTION]

Chapter 3: Problem 63

Graph the indicated functions. $$\text { Plot the graph of } f(x)=\left\\{\begin{array}{ll}3-x & \text { (for }x<1) \\\x^{2}+1 & (\text { for } x \geq 1)\end{array}\right.$$

Short Answer

Expert verified

Graph consists of a line $f(x) = 3-x$ for $x < 1$ and a parabola $f(x) = x^2 + 1$ for $x \geq 1$, meeting at $(1, 2)$.

Step by step solution

Understand the Function

The function given is a piecewise function, which means it is defined by different expressions depending on the value of the input variable, $x$. There are two parts to this function: $f(x) = 3-x$ for $x < 1$ and $f(x) = x^2 + 1$ for $x \geq 1$.

Analyze the Function for $x < 1$

For $x < 1$, the function is $f(x) = 3 - x$, which is a linear function. The slope of this line is $-1$ and the y-intercept is $3$. This line will be plotted on the graph to the left of $x = 1$, where $x$ is less than 1.

Analyze the Function for $x \geq 1$

For $x \geq 1$, the function is $f(x) = x^2 + 1$, which is a quadratic function. The graph of this part is a parabola that opens upwards with its vertex at the point $ (1, 2)$. It will be plotted for $x$ values greater than or equal to 1, including $x = 1$.

Determine the Intersection

Find the point where the two pieces meet, which occurs at $x = 1$. Evaluate both parts: at $x = 1$, $f(x) = 3 - 1 = 2$ and $f(x) = 1^2 + 1 = 2$. Since both are equal, the function is continuous at this point.

Sketch the Graph

Now combine the analyses to sketch the graph:- For $x < 1$, draw the line $f(x) = 3 - x$ until $x = 1$.- For $x \geq 1$, draw the parabola $f(x) = x^2 + 1$ starting from $x = 1$.- Make sure that at $x = 1$, both parts meet at the point $(1, 2)$, indicating continuity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions

A linear function is one of the simplest types of functions that you'll encounter in mathematics. It's a function that graphs to a straight line. The general form of a linear function is given by the equation:

$ f(x) = mx + b $ where:

$ m $ is the slope of the line, indicating how steep the line is.
$ b $ is the y-intercept, the point where the line crosses the y-axis.

For example, in the piecewise function we have $ f(x) = 3 - x $ for $ x < 1 $. Consider:

The slope $ m $ is , which means the line falls as you move from left to right.
The y-intercept $ b $ is 3, so the line crosses the y-axis at the point $ (0, 3) $.

This straight line continues until $ x = 1 $, where another function takes over.

Quadratic Functions

Quadratic functions represent a class of functions that graph to the shape of a parabola. A typical quadratic function follows the form:

$ f(x) = ax^2 + bx + c $ where:

$ a $ determines the direction and width of the parabola.
- If $ a > 0 $, the parabola opens upwards.
- If $ a < 0 $, it opens downwards.
$ b $ and $ c $ affect the position and shape of the graph.

Within our piecewise function for $ x \geq 1 $, we have $ f(x) = x^2 + 1 $. In this case:

$ a = 1 $ (positive, so the parabola opens upwards).
The vertex of this parabola occurs at $ x = 1 $, producing the point $ (1, 2) $.

This part of the piecewise function creates a smooth upward-opening curve starting at $ x = 1 $.

Graphing Functions

Graphing functions can initially seem daunting, especially when dealing with piecewise functions. However, by breaking it down into sections, it becomes much clearer. To graph a piecewise function:

Identify each section of the function, understanding its type "linear, quadratic, etc.".
Determine the domain of each section (where each part is valid).
Compute points, especially points of change where the function definition switches "like the intersection of pieces".
Plot these key points onto a coordinate plane.
Draw each segment, keeping note of open or closed circles at end points based on inequalities.

In our case, we drew a straight line from $ x < 1 $ and ensured it's continuous by joining it with a parabola starting from $ x \geq 1 $. The transition is smooth due to the matching y-values of both parts at $ x = 1 $.

Continuity of Functions

Continuity in functions is crucial, especially when graphing piecewise functions. A function is continuous at a point $ x = c $ if:

The function is defined at $ c $.
The limit of the function as $ x $ approaches $ c $ from both directions is the same.
The limit equals the function's value at $ c $.

In the current example, continuity is established at the junction $ x = 1 $. For continuity:

Evaluate the left side: $ 3 - 1 = 2 $.
Evaluate the right side: $ 1^2 + 1 = 2 $.
As both expressions equal 2, the graph smoothly connects, showing no jumps or breaks.

The result is a seamless transition between the linear and quadratic parts of the piecewise graph.

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91影视

Graph the indicated functions. $$\text { Plot the graph of } f(x)=\left\\{\begin{array}{ll}3-x & \text { (for }x<1) \\\x^{2}+1 & (\text { for } x \geq 1)\end{array}\right.$$

Short Answer

Step by step solution

Understand the Function

Analyze the Function for \(x < 1\)

Analyze the Function for \(x \geq 1\)

Determine the Intersection

Sketch the Graph

Key Concepts

Linear Functions

Quadratic Functions

Graphing Functions

Continuity of Functions

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