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Chapter 9: Problem 32

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

Short Answer

Expert verified
\[ f(x) = x^2 + 2x + 1 \]

Step by step solution

01

Identify the given function

The given function is: \[ f(x) = (x + 1)^2 \]
02

Expand the expression

Expand the binomial expression \[ (x + 1)^2 \] using the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \]. Here, \( a = x \) and \( b = 1 \), so: \[ (x + 1)^2 = x^2 + 2(x)(1) + 1^2 \]
03

Simplify the expression

Simplify the expanded expression: \[ x^2 + 2x + 1 \]

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Key Concepts

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

binomial expansion
Binomial expansion is all about expanding expressions raised to a power. In our exercise, the binomial expression is \( (x + 1)^2 \). To expand this, we use the binomial theorem.
For squares, the formula \( (a + b)^2 = a^2 + 2ab + b^2 \)鈥攄erived from the binomial theorem鈥攈elps simplify the process.
When applying this to \( (x + 1)^2 \), set \( a = x \) and \( b = 1 \), which then gives us \( x^2 + 2(x)(1) + 1^2 \).
This method works similarly for higher powers but involves more terms.
polynomial expressions
A polynomial is an expression comprising variables and coefficients. These terms are combined using addition, subtraction, and multiplication techniques.
Techniques for handling polynomials include addition, subtraction, multiplication, and constructing polynomial equations.

The expanded form \( x^2 + 2x + 1 \) from our exercise is a polynomial of degree 2, known as a quadratic polynomial.
It's constructed from the terms we expanded in the previous section. Each term contributes to the characteristic shape of the polynomial graph.
algebraic simplification
Algebraic simplification involves reducing expressions to their simplest form.
In our example, after expanding \( (x + 1)^2 \), we attained \[ x^2 + 2x + 1 \].
The goal here was to have no like terms or unnecessary parentheses remaining.
Remember, simplifying involves combining like terms and ensuring the expression is as concise as possible. Thus improving clarity and making further manipulations easier.
quadratic functions
Quadratic functions take the form \[ ax^2 + bx + c \], where \ a, b \, and \ c \ are constants.
The function from our exercise, \( f(x) = x^2 + 2x + 1 \), is a perfect example of a quadratic function.

These functions produce parabolic graphs that open upwards if \ a \ is positive and downwards if \ a \ is negative. Quadratic functions have a plethora of applications in real-world scenarios, such as projectile motion and area calculations.
Understanding and identifying the standard form is crucial in solving and graphing these functions efficiently.

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Most popular questions from this chapter

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. $$ y=|x| $$

A function h is given. Find functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x) .\) Many such pairs of functions exist. See Example 7. $$ h(x)=4(2 x-3)^{3}+(2 x-3)+5 $$

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. $$ f(x)=\frac{-1}{x^{2}+9} $$

For each pair of functions \(f\) and \(g\), find \((a) f+g,\) (b) \(f-g,\) (c) \(f g\), and \((d) \frac{f}{g}\). Give the domain for each. See Example 2. $$ f(x)=-2 x+9, \quad g(x)=-5 x+2 $$

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. $$ y=|x|+1 $$
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