/*! 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 15 If \(f(x)=x^{2}-2 x,\) find \(f(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 15

If \(f(x)=x^{2}-2 x,\) find \(f(a+1)\) and \(f(a+2).\)

Short Answer

Expert verified
For \( f(a+1) = a^2 - 1 \) and \( f(a+2) = a^2 + 2a \).

Step by step solution

01

Understand the Problem

The function given is a quadratic function: \[ f(x) = x^2 - 2x \] The task is to find the function values of \(f\) at \(a+1\) and \(a+2\).
02

Find \(f(a+1)\)

Substitute \(x = a+1\) into the function: \[ f(a+1) = (a+1)^2 - 2(a+1) \] Simplify the expression: \[ f(a+1) = a^2 + 2a + 1 - 2a - 2 \] Further simplify to get: \[ f(a+1) = a^2 - 1 \]
03

Find \(f(a+2)\)

Substitute \(x = a+2\) into the function: \[ f(a+2) = (a+2)^2 - 2(a+2) \] Simplify the expression: \[ f(a+2) = a^2 + 4a + 4 - 2a - 4 \] Further simplify to get: \[ f(a+2) = a^2 + 2a \]

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.

Function Evaluation
Evaluating a function means finding the value of the function at a certain point. For the function given in the exercise, which is a quadratic function, we need to evaluate it at specific inputs: \(a+1\) and \(a+2\).
Follow these simple steps to evaluate any function:
  • Identify the input value
  • Substitute this value into the function
  • Simplify the resulting expression
For example, for the input \(a+1\), we substitute \(x = a+1\) into the function:
Polynomial Expressions
Polynomial expressions are mathematical expressions that consist of variables and coefficients. They involve operations like addition, subtraction, and multiplication. In our exercise, the given function, \(f(x) = x^2 - 2x\), is a quadratic polynomial.
Let's breakdown the main components of this polynomial expression:
  • The term \(x^2\) is a second-degree term since the variable is squared.
  • The term \(-2x\) is a first-degree term, which means the variable is raised to the power of one.
Polynomials are often used to model various real-world situations, making them very important in algebra.
Substitution
Substitution is a method used in algebra to simplify expressions or solve equations. It involves replacing a variable with a given value or another expression. In our exercise, we use substitution to find the values of the function at \(a+1\) and \(a+2\).
Here鈥檚 how you can use substitution to simplify an equation:
  • Identify the value that you have to substitute. For instance, \(a+1\) or \(a+2\).
  • Replace every instance of the variable in the original function with this value.
  • Simplify the resulting expression by carrying out algebraic operations like expanding, combining like terms, and reducing.
For example, when substituting \(x = a+1\) in the function \(f(x) = x^2 - 2x\), the expression becomes \(f(a+1) = (a+1)^2 - 2(a+1)\), which simplifies to \(a^2 - 1\).

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

Find the points of intersection of the graphs of the functions. (Use the specified viewing window.) $$f(x)=2 x-1 ; g(x)=x^{2}-2 ;[-4,4] \text { by }[-6,10]$$

The expressions may be factored as shown. Find the missing factors. $$x^{-1 / 4}+6 x^{1 / 4}=x^{-1 / 4}(\quad)$$

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. $$g\left(\frac{1}{u}\right)$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$x^{-3} \cdot x^{7}$$

A store estimates that the total revenue (in dollars) from the sale of \(x\) bicycles per year is given by the function \(R(x)=250 x-.2 x^{2}.\) (a) Graph \(R(x)\) in the window $$[200,500] \text {by } [42000,75000].$$ (b) What sales level produces a revenue of \(\$ 63,000 ?\) (c) What revenue is received from the sale of 400 bicycles? (d) Consider the situation of part (c). If the sales level were to decrease by 50 bicycles, by how much would revenue fail? (c) The store believes that, if it spends \(\$5000\) in advertising, it can raise the total sales from 400 to 450 bicycles next year. Should it spend the $5000? Explain your conclusion.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.