/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 $$ f(x)=(x+1)^{2}+1 $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 36

$$ f(x)=(x+1)^{2}+1 $$

Short Answer

Expert verified
The simplified function is \( f(x) = x^2 + 2x + 2 \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = (x+1)^2 + 1 \). This is a quadratic function with a vertical shift.
02

Apply the Binomial Expansion

Expand the term \( (x+1)^2 \) using the binomial theorem: \( (x+1)^2 = x^2 + 2x + 1 \).
03

Combine Like Terms

Add the constant term from the function definition: \( f(x) = x^2 + 2x + 1 + 1 = x^2 + 2x + 2 \).
04

Final Function

The simplified quadratic function is now: \( f(x) = x^2 + 2x + 2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Binomial Expansion
In this exercise, we start with the function \( f(x) = (x+1)^2 + 1 \). To simplify this function, we need to expand the binomial part. The binomial theorem helps us expand expressions that are raised to a power.

When we have \( (a + b)^n \), for example, the theorem tells us that we can expand it as: \[ (a + b)^2 = a^2 + 2ab + b^2 \]

In this function, \((x+1)^2\), both \(a \) and \(b \) are \(1 \). By applying the binomial theorem:
  • \(a = x \)
  • \(b = 1 \)
  • \(n = 2 \)


The expanded form becomes: \[ (x + 1)^2 = x^2 + 2x + 1 \]

Now we have simplified the quadratic part of the function.
Function Transformation
Function transformation involves shifting or stretching a function to create a new one. The original function given is \[ f(x) = (x + 1)^2 + 1 \]

After expanding the quadratic term (or the binomial), we have: \[ (x + 1)^2 = x^2 + 2x + 1 \]

Adding the last constant \( +1 \) from the original function:
  • \( (x + 1)^2 + 1 = x^2 + 2x + 1 + 1 \)
    • This results in: \[ x^2 + 2x + 2 \]

      The transformation is a vertical shift of one unit up. This is why we added the constant \( +1 \).
    • The original function has been transformed from the vertex form into standard form.


    This step ensures we understand how to manipulate the components of a function to get a new shape.
Combining Like Terms
When working with algebraic expressions, it鈥檚 crucial to combine like terms to simplify the expression. After expanding the binomial term, we had:
\[ (x + 1)^2 = x^2 + 2x + 1 \]

Adding the remaining \( +1 \) gives: \[ x^2 + 2x + 1 + 1 = x^2 + 2x + 2 \]

Like terms are terms that have the same variables raised to the same power. For instance, in \( x^2 + 2x + 1 \), the terms \(x^2\), \(2x\), and \(1\) are all like terms already somewhat combined. Only constants are combined in this case:
  • \( 1 + 1 = 2 \)


    • Combining constants helps reach the finalized expression. This process simplifies the function so we can easily read and understand it.

    The final function becomes: \[ f(x) = x^2 + 2x + 2 \]

    Combining like terms is a step that can鈥檛 be skipped, as it ensures the polynomial is in its simplest and most understandable form.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Compared to the graph of \(f(x)=\sqrt{x}\), the graph of \(g(x)=\frac{1}{2} \sqrt{x}\) is (stretched / shrunken) by a factor of ______. Some points on the graph of \(f(x)\) are (0,0),(4,2) , and (16,4) . Corresponding points on the graph of \(g(x)\) are (0,0) (4, ____), and (16, ____).

Graph each piecewise linear function. \(f(x)=\left\\{\begin{array}{ll}-2 x & \text { if } x<-3 \\ 3 x-1 & \text { if }-3 \leq x \leq 2 \\ -4 x & \text { if } x>2\end{array}\right.\)

Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ y=4 x-x^{2} $$

Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ y=5 x $$

A function \(f\) is an even function if \(f(-x)=f(x)\) for all \(x\) in the domain of \(f\). A function \(f\) is an odd function if \(f(-x)=-f(x)\) for all \(x\) in the domain of \(f\). To see how these ideas relate to symmetry, work in order. Use the preceding definition to determine whether the function \(f(x)=x^{n}\) is an even function or an odd function for \(n=2, n=4,\) and \(n=6\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.