/*! 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 73 Find \(g(w)\) and \(g(w+1)\). ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 73

Find \(g(w)\) and \(g(w+1)\). \(g(x)=3 x-5\)

Short Answer

Expert verified
\(g(w) = 3w - 5\); \(g(w+1) = 3w - 2\).

Step by step solution

01

Substitute for g(w)

To find \(g(w)\), substitute \(x = w\) into the function. This gives us \(g(w) = 3w - 5\).
02

Substitute for g(w+1)

To find \(g(w+1)\), substitute \(x = w+1\) into the function. This gives us \(g(w+1) = 3(w+1) - 5\).
03

Simplify g(w+1)

Simplify the expression for \(g(w+1)\) by distributing the 3 and combining like terms: \(g(w+1) = 3w + 3 - 5\).
04

Combine Like Terms

Combine the constant terms in \(g(w+1) = 3w + 3 - 5\): \(g(w+1) = 3w - 2\).

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.

Function Substitution
Function substitution is a process that involves replacing the input of a function with a specified value or expression. This allows us to tailor functions to specific problems or evaluations. In the context of our exercise, the function is initially given as:\[ g(x) = 3x - 5 \]We needed to find two functions: \( g(w) \) and \( g(w+1) \). To do this, we simply substitute the variables:
  • For \( g(w) \), substitute \( x \) with \( w \).
  • For \( g(w+1) \), substitute \( x \) with \( w+1 \).
This substitution process helps us to translate general forms into specific forms based on the requirements, allowing us to analyze or manipulate the function for further simplification or evaluation.
Simplifying Expressions
Once we perform function substitution, the next step is usually to simplify the resulting expression. Simplification makes an expression easier to understand or use. For example:After substituting into \( g(w+1) \), we obtained:\[ g(w+1) = 3(w+1) - 5 \]To simplify, we need to perform operations like distribution. This involves multiplying the number in front of the parentheses by each term inside:\[ 3(w+1) = 3w + 3 \]Once distributed, the expression becomes:\[ g(w+1) = 3w + 3 - 5 \]Simplifying an expression is a key part of solving problems in algebra, as it often uncovers the most compact and usable form of the expression.
Combining Like Terms
Combining like terms is a crucial step in algebra to tidy up and simplify expressions further after substitution and distribution. Like terms are terms that contain the same variables raised to the same power, and constants can also be combined.For example, in the expression:\[ g(w+1) = 3w + 3 - 5 \]Here, the terms \( 3 \) and \( -5 \) are constants and can be combined:\[ 3 - 5 = -2 \]So the simplified expression becomes:\[ g(w+1) = 3w - 2 \]Combining like terms reduces the complexity of expressions and helps make calculations more straightforward, which is especially useful in solving equations and analyzing mathematical models.

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

Complete table. \(f(r)=-2 r^{2}+1\) \(\begin{array}{|r|l|}\hline \text { Input } & \text { Output } \\\\\hline-1.7 & \\\0.9 & \\\5.4 & \\\\\hline\end{array}\)

Find \(g(2)\) and \(g(3)\). \(g(x)=x^{3}-x\)

Graph function. \(f(x)=-\frac{3}{2} x-3\)

Write an equation for a linear function whose graph has the given characteristics. Slope \(\frac{1}{5},\) passes through \((10,1)\)

Write an equation for a linear function whose graph has the given characteristics. Passes through \((1,2),\) perpendicular to the graph of \(g(x)=-\frac{x}{6}+1\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

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.