/*! 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 31 If \(f(x)=x^{2},\) find \(f(x+h)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 31

If \(f(x)=x^{2},\) find \(f(x+h)-f(x)\) and simplify.

Short Answer

Expert verified
The simplified expression is \(2xh + h^2\).

Step by step solution

01

Understand the problem

Given the function \(f(x) = x^2\), find the expression for \(f(x + h) - f(x)\).
02

Plug in \(x + h\) into the function

Replace \(x\) with \(x + h\) in the function \(f(x)\): \[f(x + h) = (x + h)^2\]
03

Expand \((x + h)^2\)

Expand the expression using the binomial theorem: \[f(x + h) = (x + h)^2 = x^2 + 2xh + h^2\]
04

Subtract \(f(x)\) from \(f(x + h)\)

Now compute \(f(x + h) - f(x)\): \[f(x + h) - f(x) = (x^2 + 2xh + h^2) - x^2\]
05

Simplify the expression

Combine like terms and simplify: \[f(x + h) - f(x) = x^2 + 2xh + h^2 - x^2 = 2xh + h^2\]

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 Theorem
The binomial theorem is a fundamental concept used in algebra for expanding expressions that are raised to a power. It states that:

  • For any non-negative integer `n`,

'x + h)^2'
To expand When you see an expression such as
<>

Simplification
Simplifying algebraic expressions involves reducing them to their simplest form. This process helps in understanding and solving the expression more easily.
<>
  • First, substitute the value or expression given into the function or equation.
  • Next, collect and combine like terms. For instance, terms involving the same variable and exponent.
  • Finally, eliminate any unnecessary terms to achieve the simplified form.
Simplification is crucial in making the problem more manageable. Always aim to end up with the most reduced form of the equation or expression. By simplifying step-by-step, you ensure clarity and avoid errors.

In our example of finding `f(x + h) - f(x)` for the function `f(x) = x^2`, simplification was employed to reduce the expanded polynomial to

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

Evaluate \(f(4)\). $$f(x)=x^{-1 / 2}$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$f(f(x))$$

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x},\) and \(h(x)=\frac{x+1}{3 x-1} .\) Express the following as rational functions. $$g\left(\frac{1}{u}\right)$$

Calculate the compound amount from the given data. principal \(=\$ 500,\) compounded annually, 6 years, annual rate \(=6 \%\)

Let \(f(x)=x^{6}, g(x)=\frac{x}{1-x},\) and \(h(x)=x^{3}-5 x^{2}+1 .\) Calculate the following functions. $$f(g(x))$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

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.