/*! 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 81 Function Notation Given \(f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 81

Function Notation Given \(f(x)=x^{2}+x,\) find and simplify the following. a) \(f(a)+f(-a)\) b) \(f(a+h)\) c) \(f(a+h)-f(a)\)

Short Answer

Expert verified
a) 2a^2; b) a^2 + 2ah + h^2 + a + h; c) 2ah + h^2 + h

Step by step solution

01

Expression for f(a)

First, find the expression for \(f(a)\). Substitute \(a\) into \(f(x)\), so \(f(a) = a^2 + a\).
02

Expression for f(-a)

Now, find the expression for \(f(-a)\). Substitute \(-a\) into \(f(x)\), so \(f(-a) = (-a)^2 + (-a) = a^2 - a\).
03

Sum f(a) + f(-a)

Add the expressions determined in the previous steps: \(f(a) + f(-a) = (a^2 + a) + (a^2 - a) = 2a^2\).
04

Expression for f(a+h)

Find the expression for \(f(a+h)\). Substitute \(a+h\) into \(f(x)\), so \(f(a+h) = (a + h)^2 + (a + h)\).
05

Simplify f(a+h)

Expand and simplify the expression for \(f(a+h)\): \(f(a+h) = a^2 + 2ah + h^2 + a + h\).
06

Expression for f(a)

Recall the expression for \(f(a)\): \(f(a) = a^2 + a\).
07

Difference f(a+h) - f(a)

Subtract \(f(a)\) from \(f(a+h)\): \(f(a+h) - f(a) = (a^2 + 2ah + h^2 + a + h) - (a^2 + a)\).
08

Simplify the difference

Simplify the expression: \(f(a+h) - f(a) = a^2 + 2ah + h^2 + a + h - a^2 - a = 2ah + h^2 + h\).

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
Function evaluation is the process of finding the value of a function for a given input. In simpler terms, it means substituting a given number into the function in place of the variable and then calculating the result. For example, if we have a function \(f(x) = x^2 + x\), and we want to find the value of \(f(a)\), we substitute \(a\) for \(x\) in the function, resulting in \(f(a) = a^2 + a\).
Let's see how evaluation works with different values:
  • For \(a\): \(f(a) = a^2 + a\)
  • For \(-a\): \(f(-a) = (-a)^2 + (-a) = a^2 - a\)
By evaluating functions for different values, we can better understand how the function behaves under various conditions.
function operations
Function operations involve combining two or more functions in various ways, such as adding, subtracting, multiplying, or dividing them. In this exercise, we focus on addition and subtraction:
Let's combine \(f(a)\) and \(f(-a)\):
  • First, find \(f(a)\) and \(f(-a)\) as we did in the previous section.
  • Next, add them: \(f(a) + f(-a) = (a^2 + a) + (a^2 - a) = a^2 + a + a^2 - a = 2a^2\).
This shows that the positive and negative parts of the function cancel each other out, leaving a simplified form.
For the subtraction operation:
  • We find \(f(a+h) - f(a)\).
  • This involves substituting \(a+h\) into the function: \(f(a+h) = (a+h)^2 + (a+h) = a^2 + 2ah + h^2 + a + h\).
  • Next, subtract \(f(a)\): \(f(a+h) - f(a) = (a^2 + 2ah + h^2 + a + h) - (a^2 + a) = 2ah + h^2 + h\).
By performing these operations, we learn how functions interact with each other and uncover useful patterns.
simplification
Simplification is the process of reducing a complex expression into a more manageable and understandable form. Let's simplify some expressions step-by-step:
First, let's simplify \(f(a+h)\):
  • Start with its expanded form: \(f(a+h) = (a+h)^2 + (a+h) = a^2 + 2ah + h^2 + a + h\).
  • Combine like terms: \(a^2 + a, 2ah, h^2, h\).
The simplified form is \(a^2 + 2ah + h^2 + a + h\).
Now let's simplify the difference \(f(a+h) - f(a)\):
  • Write the expressions: \(f(a+h) = a^2 + 2ah + h^2 + a + h \) and \(f(a) = a^2 + a\).
  • Subtract \(f(a)\) from \(f(a+h)\): \(f(a+h) - f(a) = a^2 + 2ah + h^2 + a + h - a^2 - a\).
  • Cancel out the common terms: \(a^2 - a^2\) and \(a - a\).
The simplified result is \(2ah + h^2 + h\).

Through simplification, we make expressions cleaner and easier to work with, revealing the essential components of the equation.

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

Solve. Fireworks Displays. Fireworks are typically launched from a mortar with an upward velocity (initial speed) of about \(64 \mathrm{ft} / \mathrm{sec} .\) The height \(h(t)\) in feet, of a "weeping willow" display, \(t\) seconds after having been launched from an 80 -ft high rooftop, is given by $$h(t)=-16 t^{2}+64 t+80 $$ How long will it take the cardboard shell from the fireworks to reach the ground? (image cannot copy) (graph cannot copy)

Explain how one could construct a polynomial with four terms that can be factored by grouping three terms together.

If the graph of \(f(x)=a x^{2}+b x+c\) has no \(x\) -intercepts, what can you conclude about the equation \(a x^{2}+b x+c=0 ?\)

Solve. If each side of a square is lengthened by \(6 \mathrm{cm},\) the area becomes \(144 \mathrm{cm}^{2} .\) Find the length of a side of the original square.

Solve. Furniture. The base of a triangular tabletop is 20 in. longer than the height. The area is 750 in ". Find the height and the base. (image cannot copy)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.