/*! 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 54 If \(f(x)=3 x+3, g(x)=4 x^{2}-6 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 54

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7\) find the following. $$ f(-1) $$

Short Answer

Expert verified
\( f(-1) = 0 \)

Step by step solution

01

Identify the Function

The function we need to use to find the value of \( f(-1) \) is \( f(x) = 3x + 3 \).
02

Substitute the Value into the Function

Substitute \( x = -1 \) into the function \( f(x) = 3x + 3 \). This gives us: \( f(-1) = 3(-1) + 3 \).
03

Simplify the Calculation

Simplify the expression \( 3(-1) + 3 \). This results in \(-3 + 3\).
04

Calculate the Final Value

Compute the final value of the simplified expression, \(-3 + 3 = 0\). So, \( f(-1) = 0 \).

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.

Substitution
Function evaluation often starts with substitution. Substitution is when you replace a variable in an equation with a specific value. In our exercise, we are evaluating the function \(f(x) = 3x + 3\) at \(x = -1\). That means replacing \(x\) with \(-1\) in the function.The steps are straightforward:
  • Find the variable in the expression.
  • Replace the variable with the specified number.
In this specific exercise, the substitution gives us \(f(-1)\) which translates the equation to \(3(-1) + 3\). By performing substitution correctly, we set the stage for further simplifications.
Substitution is crucial because it moves the problem from an abstract formula to a concrete calculation, making it easier to resolve.
Algebraic Simplification
After we substitute a value into the function, the next step is algebraic simplification. Simplification is the process of reducing an expression to its simplest form.In our example:
  • Start with \(3(-1) + 3\).
  • Calculate \(3 \times -1 = -3\).
  • Then, compute \(-3 + 3\). The result of this is \(0\).
Algebraic simplification is essential because it allows for a clearer and easier solution by reducing complexities. The focus is on step-by-step reduction until the simplest form is achieved, which leads directly to the final result of the function evaluation.
Polynomials
In mathematics, polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. Our function \(f(x) = 3x + 3\) is a simple example of a polynomial function.Polynomials are characterized by:
  • Terms: The building blocks of polynomials, each term is a product of coefficients and variables raised to a non-negative integer power.
  • Degree: The highest power of the variable in the polynomial. For \(f(x) = 3x + 3\), the degree is 1 as the highest power of \(x\) is 1.
  • Operations: You can perform addition, subtraction, and especially multiplication on polynomials to combine them in various ways.
Understanding polynomials help in interpreting the behaviors of functions like growth, increases, or decreases, which can be observed through their structure. For learners, tackling polynomials facilitates understanding how different functions are solved and evaluated.

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

The function \(V(x)=x^{3}\) may be used to find the volume of a cube if we are given the length \(x\) of a side. Find the volume of a cube whose side is 14 inches.

Suppose that \(y=f(x)\) and it is true that \(f(7)=50 .\) Determine whether each is true or false. See the third Concept Check in this section. A possible function is \(f(x)=10 x-20\)

In your own words, explain how to find \(x\) - and \(y\) -intercepts.

While manufacturing two different digital camera models, Kodak found that the basic model costs 55 dollars to produce, whereas the deluxe model costs 75 dollars. The weekly budget for these two models is limited to 33,000 dollars in production costs. The linear equation that models this situation is \(55 x+75 y=33,000,\) where \(x\) represents the number of basic models and \(y\) the number of deluxe models. a. Complete the ordered pair solution \((0,)\) of this equation. Describe the manufacturing situation this solution corresponds to. b. Complete the ordered pair solution \((, 0)\) of this equation. Describe the manufacturing situation this solution corresponds to. c. If 350 deluxe models are produced, find the greatest number of basic models that can be made in one week.

The amount of money (in billions of dollars) spent by the Boeing Company and subsidiaries on research and development annually is represented by the function \(R(x)=0.382 x+2.21,\) where \(x\) is the number of years after \(2005 .\) (Source: Boeing Corporation) a. Find and interpret \(R(2)\) b. Estimate the amount of money spent on research and development by Boeing in 2014
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.