/*! 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 11 Let \(f(x)=-3 x+4\) and \(g(x)=-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 11

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(3) $$

Short Answer

Expert verified
4

Step by step solution

01

Identifying the Function

Firstly, identify the function we need to evaluate. In this case, we're asked to evaluate the function \( g(x) \) at \( x = 3 \). The given function is \( g(x) = -x^2 + 4x + 1 \).
02

Substitute \( x \) with 3

Next, substitute \( x \) with 3 in the function \( g(x) \). This means we need to find \( g(3) \). This step transforms the expression to \( g(3) = -3^2 + 4(3) + 1 \).
03

Perform Squaring and Multiplication

Calculate the squaring and multiplication operations first. For the squaring: \( -3^2 = -9 \). Then for the multiplication: \( 4(3) = 12 \). We now have \( g(3) = -9 + 12 + 1 \).
04

Combining the Terms

Finally, combine all the terms in the equation: \( -9 + 12 + 1 \). Simplify this to get the result: \( 4 \). Therefore, we find \( g(3) = 4 \).

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
When evaluating functions, substitution is a key method. Substitution means replacing a variable with a given value. For example, in the exercise, we need to find the value of the function \( g(x) \) at \( x = 3 \). Initially, the function is expressed as \( g(x) = -x^2 + 4x + 1 \). By substituting \( x \) with \( 3 \), the function becomes \( g(3) \). This results in \( g(3) = -3^2 + 4(3) + 1 \). Here, \( 3 \) replaces every occurrence of \( x \) in the original expression.
Simplification
Simplification is the process of reducing an expression to its simplest form. After substituting the value for \( x \), the next step is to simplify the expression. This involves performing mathematical operations like squaring, multiplication, and addition or subtraction. Starting with \( g(3) = -3^2 + 4(3) + 1 \), first, we calculate the square of 3 which is 9, remembering the negative sign: \( -3^2 = -9 \). Next, we perform the multiplication: \( 4(3) = 12 \). Combine these results along with the constant 1: \( -9 + 12 + 1 \). Adding these gives \( 4 \). Therefore, \( g(3) = 4 \). By following these steps, we simplify the expression to find the value.
Polynomial Functions
Polynomial functions are algebraic expressions that include variables raised to whole number exponents and constants. The general form is \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \). In the given exercise, the function \( g(x) = -x^2 + 4x + 1 \) is a polynomial of degree 2. This means the highest exponent of the variable \( x \) is 2. Polynomial functions can be quadratic, linear, or of higher degrees. Evaluating them involves substituting the variable and simplifying, as shown in our step-by-step solution. Understanding polynomial functions is fundamental in algebra because they appear frequently in various mathematical problems.

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

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(x+h) $$

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ f=\\{(-2,2),(-1,-1),(2,-1)\\} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x+1=0\) and \(y-2=0\)

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(t) $$

Graph each linear or constant function. Give the domain and range. $$ f(x)=0 $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.