**Title: Understanding Piecewise Functions and Limits**---**Introduction**In this section, we will explore the concept of piecewise functions and their graphical representation. We will also look into how to evaluate the limit of such functions as they approach a specific point.---**Piecewise Function Example**Consider the following piecewise function:\[f(x) = \begin{cases} x + 5, & \text{if } x \leq 3 \\ -2x - 1, & \text{if } x > 3 \end{cases}\]**Task**We are tasked with sketching the graph of this function and evaluating the limit of $ f(x) $ as $ x $ approaches 3.---**Graphical Representation**- **Part 1**: For $ x \leq 3 $, the function is $ f(x) = x + 5 $. - This is a straight line with a slope of 1 and a y-intercept at 5. - The line continues until $ x = 3 $, where it is represented as a solid dot indicating that the point is included in this segment.- **Part 2**: For $ x > 3 $, the function becomes $ f(x) = -2x - 1 $. - This is a line with a slope of -2 and a y-intercept at -1. - The graph begins at $ x = 3 $, shown as an open circle to indicate that this point is not included.**Graph Detail Explanation**- The graph shows two distinct linear parts for the function.- As $ x $ increases up to 3, the line follows the equation $ y = x + 5 $, reaching a value of 8 at $ x = 3 $.- Beyond $ x = 3 $, the line shifts to follow $ y = -2x - 1 $, beginning at the open circle above $ x = 3 $.---**Evaluating the Limit**To find $\lim_{x \to 3} f(x)$, we need to consider the behavior of the function as $ x $ approaches 3 from both sides:- As $ x $ approaches 3 from the left ($ x \leq 3 $), $ f(x) = x + 5 $ approaches

Given, f(x)=x+5, if x≤-3-2x-1, if x>-3

Answered: Sketch the graph of the function f and…

Math

Calculus

Sketch the graph of the function f and evaluate lim f(x). (x+5, if x s-3 f(x) = %3D -2x-1, ifx >-3

Intermediate Algebra

10th Edition

ISBN:9781285195728

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter9: Functions

Section9.1: Relations And Functions

Problem 75PS

See similar textbooks

Related questions

Q: Sketch the following function. Also, make sure to clearly label the point that corresponds to x = 2…

A: Given that the function f(x)=-3x-2+1. Sketch the given function and to label the point…

Q: S 2x2 + 3x – 3 if x -1 1) Complete an Xy-chart for each piece of the given function. Be sure to use…

Q: Write the function in its final position. f(x)= x³ first moved 2 units downward, then 3 units to the…

A: The objective is to write the function in its final position.

Q: Find f(a), f(a+h), and the difference quotient for the function given below.

Q: x, if -1くx<

Q: If f(4) = 10 then the point (4, ) is on the graph of f.

A: To find: If f(4) = 10 then the point (4, _____) is on the graph of f.

Q: If ƒ(a - x) = ƒ(a + x), show that g(x) = ƒ(x + a) is an even function.

A: Given that ƒ(a - x) = ƒ(a + x)To show that g(x) = ƒ(x + a) is an even function.

Q: 3- if x 2 -2 *(x) = —х + 1 2 1- -4 -2 -1 1 2 3 4 -1 -2- -3 2.

A: we have to draw the graph

Q: Display the graph of the function f (x) = x2 from the set of integers to the set of integers.

Q: (x(x+1), if x -2 f(x) ={ 4x+5, if -2くxく1 - 2x +3, if x 21

Q: Give the range of the function in interval notation f(x) =2(x+1)2 + 7

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Title: Understanding Piecewise Functions and Limits**

---

**Introduction**

In this section, we will explore the concept of piecewise functions and their graphical representation. We will also look into how to evaluate the limit of such functions as they approach a specific point.

---

**Piecewise Function Example**

Consider the following piecewise function:

\[
f(x) =
\begin{cases}
x + 5, & \text{if } x \leq 3 \\
-2x - 1, & \text{if } x > 3
\end{cases}
\]

**Task**

We are tasked with sketching the graph of this function and evaluating the limit of $ f(x) $ as $ x $ approaches 3.

---

**Graphical Representation**

- **Part 1**: For $ x \leq 3 $, the function is $ f(x) = x + 5 $.
- This is a straight line with a slope of 1 and a y-intercept at 5.
- The line continues until $ x = 3 $, where it is represented as a solid dot indicating that the point is included in this segment.

- **Part 2**: For $ x > 3 $, the function becomes $ f(x) = -2x - 1 $.
- This is a line with a slope of -2 and a y-intercept at -1.
- The graph begins at $ x = 3 $, shown as an open circle to indicate that this point is not included.

**Graph Detail Explanation**

- The graph shows two distinct linear parts for the function.
- As $ x $ increases up to 3, the line follows the equation $ y = x + 5 $, reaching a value of 8 at $ x = 3 $.
- Beyond $ x = 3 $, the line shifts to follow $ y = -2x - 1 $, beginning at the open circle above $ x = 3 $.

---

**Evaluating the Limit**

To find $\lim_{x \to 3} f(x)$, we need to consider the behavior of the function as $ x $ approaches 3 from both sides:

- As $ x $ approaches 3 from the left ($ x \leq 3 $), $ f(x) = x + 5 $ approaches

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Limits and Continuity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Why does the vertical line test tell as whether the graph of a relation represents a function?
Why does the horizontal line test tell us whether the graph of a function is one-to-one?
Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?
How do we determine the domain of a function defined by an equation?
Suppose the range of a function fis [5,8] . What is the range of |f(x)| ?
In these problem you are asked to find a function that models a real-life situation and then use the model to answer questions about the situation. Use the guidelines on page 273 to help you. Minimizing a Sum Find two possible numbers whose sum is 100 and the sum or whose squares is a minimum.
If a function f is an even function, then what type of symmetry does the graph of f have?
During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?
What are the limits of function values?
What are facts and characteristics of a piecewise function, What are examples and non examples of a piecewise function as well.
Briefly explain what the Mean Value Theorem tells us. Why do we require that f be a continuous function in order to apply this theorem? You may sketch a graph to explain your reasoning.
ER STATE COLLEGE My Insttution Courses Be Help MAC1105-20222-1V-M-001 College Algebra - TR 01:00 PM IN|VC > Chapter 3 - Functions and Graphs > Sec Use the given graph of f to graph the function where g(x) = f( – 1) + 4 The function f(x) is shown below. Graph the transformed function g by first translating and graphing the four given reference points of the new transformed function then click on the "line segment" button to connect the points. 7- 4 2. P Type here to search 近 3.