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Chapter 5: Problem 23

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

Short Answer

Expert verified
(F+G)(-4)=31

Step by step solution

01

- Define F(x) and G(x)

First, understand that we are given two functions: \(f(x) = -3x + 1\) and \(g(x) = x^2 + 2\). These will be combined to form \(F(x) + G(x)\).
02

- Form F(x) + G(x)

Next, add the two functions together. The resulting function is \((f+g)(x) = f(x) + g(x) = (-3x + 1) + (x^2 + 2)\). Simplify this to get \((f+g)(x) = x^2 - 3x + 3\).
03

- Evaluate at x = -4

Now substitute \(x = -4\) into the combined function.\[(f+g)(-4) = (-4)^2 - 3(-4) + 3\]
04

- Simplify

Calculate the value:\[(f+g)(-4) = 16 + 12 + 3 = 31\]. Thus, \((F+G)(-4) = 31\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

function addition
Function addition is a process where we combine two functions to create a new function. In our problem, we are given two functions:
  • $$f(x) = -3x + 1$$
  • $$g(x) = x^2 + 2$$

When we add these functions, we simply add their expressions together. Hence, \[(f + g)(x) = f(x) + g(x)\]
This results in: \[f(x) + g(x) = (-3x + 1) + (x^2 + 2)\].
This is the first step in solving the problem. Once you understand this, combining functions becomes an easy and straightforward task.
polynomial functions
A polynomial function is a mathematical expression consisting of variables (often denoted as x) raised to non-negative integer powers and their corresponding coefficients. In our example:
  • First function $$f(x) = -3x + 1$$ is a linear polynomial (degree 1).
  • Second function $$g(x) = x^2 + 2$$ is a quadratic polynomial (degree 2).

When we combine these two functions through addition, our resultant function is \[x^2 - 3x + 3\], which is a polynomial of degree 2 since the highest power of x is 2. Understanding the nature and degree of polynomial functions helps in simplifying complex expressions and finding solutions efficiently.
function evaluation
Function evaluation means finding the value of a function for a specific value of x. In our problem, we need to evaluate \( (f + g)(x) \) at \( x = -4 \). After forming our combined function \( (f + g)(x) = x^2 - 3x + 3 \), we substitute \( -4 \) for x: \[(f + g)(-4) = (-4)^2 - 3(-4) + 3\]
This step involves basic substitution followed by arithmetic operations. This entire process boils down to computing numerical values, making sure each step is carefully carried out.
simplification
Simplification involves reducing a function or expression to its simplest form. After substituting \( x = -4 \) in the combined function, we need to simplify \( (f + g)(-4) \): \[(f + g)(-4) = 16 + 12 + 3 = 31\]
Simplification ensures clarity and correctness of the result. When solving polynomial functions or any mathematical expression, breaking down each operation into simpler steps helps avoid errors and makes problem-solving much easier. Always double-check each step to confirm the accuracy of your simplification.

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