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Chapter 8: Problem 3

Let \(f(x)=3 x^{5}+x\) and \(g(x)=x^{2}-1\). a. Construct the following functions. \(j(x)=f(x)+g(x), \quad k(x)=f(x)-g(x), \quad l(x)=f(x) \cdot g(x)\) b. Evaluate \(j(2), k(3),\) and \(l(-1)\)

Short Answer

Expert verified
j(2)=101, k(3)=724, l(-1)=0

Step by step solution

01

- Define the Given Functions

The functions are given as:\(f(x) = 3x^5 + x\) and \(g(x) = x^2 - 1\)
02

- Construct \(j(x)\)

To find \(j(x)\), add \(f(x)\) and \(g(x)\):\[ j(x) = f(x) + g(x) \]\[ j(x) = (3x^5 + x) + (x^2 - 1) \]Combine like terms:\[ j(x) = 3x^5 + x^2 + x - 1 \]
03

- Construct \(k(x)\)

To find \(k(x)\), subtract \(g(x)\) from \(f(x)\):\[ k(x) = f(x) - g(x) \]\[ k(x) = (3x^5 + x) - (x^2 - 1) \]Combine like terms:\[ k(x) = 3x^5 - x^2 + x + 1 \]
04

- Construct \(l(x)\)

To find \(l(x)\), multiply \(f(x)\) and \(g(x)\):\[ l(x) = f(x) \times g(x) \]\[ l(x) = (3x^5 + x) \times (x^2 - 1) \]Use the distributive property:\[ l(x) = 3x^5(x^2 - 1) + x(x^2 - 1) \]\[ l(x) = 3x^7 - 3x^5 + x^3 - x \]
05

- Evaluate \(j(2)\)

Evaluate \(j(x)\) at \(x = 2\):\[ j(2) = 3(2)^5 + 2^2 + 2 - 1 \]Calculate the values:\[ j(2) = 3(32) + 4 + 2 - 1 \]\[ j(2) = 96 + 4 + 2 - 1 \]\[ j(2) = 101 \]
06

- Evaluate \(k(3)\)

Evaluate \(k(x)\) at \(x = 3\):\[ k(3) = 3(3)^5 - 3^2 + 3 + 1 \]Calculate the values:\[ k(3) = 3(243) - 9 + 3 + 1 \]\[ k(3) = 729 - 9 + 3 + 1 \]\[ k(3) = 724 \]
07

- Evaluate \(l(-1)\)

Evaluate \(l(x)\) at \(x = -1\):\[ l(-1) = 3(-1)^7 - 3(-1)^5 + (-1)^3 - (-1) \]Calculate the values:\[ l(-1) = 3(-1) - 3(-1) - 1 + 1 \]\[ l(-1) = -3 + 3 - 1 + 1 \]\[ l(-1) = 0 \]

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Key Concepts

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

Function Evaluation
When we talk about function evaluation, we're referring to substituting a specific value of the variable into the function and calculating the result. For instance, if you have a function like \(f(x) = 3x^5 + x\), and need to evaluate \(f(2)\), you'll replace \(x\) with 2. This process involves:
1. Substituting the given value for the variable.
2. Performing the necessary arithmetic operations.
Using our example, for \(f(2)\):
\[f(2) = 3(2)^5 + 2 = 3(32) + 2 = 96 + 2 = 98\]
It鈥檚 straightforward arithmetic, but ensure no steps are skipped to avoid mistakes.
Addition and Subtraction of Functions
Adding and subtracting functions is the process of combining them in a way that either adds or subtracts their corresponding outputs. Consider functions \(f(x)\) and \(g(x)\) like in our example:
\[f(x) = 3x^5 + x\]
\[g(x) = x^2 - 1\]
To find \(j(x) = f(x) + g(x)\), you simply add the expressions of \(f(x)\) and \(g(x)\):
\[ j(x) = (3x^5 + x) + (x^2 - 1) = 3x^5 + x^2 + x - 1\]
For subtraction like \(k(x) = f(x) - g(x)\), do the same but subtract:
\[ k(x) = (3x^5 + x) - (x^2 - 1) = 3x^5 - x^2 + x + 1\]Combining terms like this helps simplify complex problems into single functions.
Multiplication of Functions
Multiplying functions involves using each term from one function to multiply every term in the other. Given \(f(x) = 3x^5 + x\) and \(g(x) = x^2 - 1\), multiplying them to get \(l(x)\) goes like this:
\[ l(x) = f(x) \times g(x) = (3x^5 + x) \times (x^2 - 1)\]
Using distributive property, multiply each term in \(f(x)\) by each term in \(g(x)\):
\[ l(x) = 3x^5 \times x^2 + 3x^5 \times (-1) + x \times x^2 + x \times (-1) = 3x^7 - 3x^5 + x^3 - x\]
This process results in a new function formed by these multiplications.
Polynomial Manipulation
Polynomial manipulation involves taking any polynomial, like \(f(x) = 3x^5 + x\) or \(g(x) = x^2 - 1\), and performing operations on them such as addition, subtraction, or multiplication. These methods help simplify and understand complex expressions. When combining like terms, always:
- Identify similar terms having the same variable and exponent.
- Perform arithmetic operations with their coefficients.
Consider the manipulation during addition and subtraction:
\[ j(x) = 3x^5 + x^2 + x - 1 \]
\[ k(x) = 3x^5 - x^2 + x + 1 \]
In multiplication, each term affects one another as shown in \[ l(x) = 3x^7 - 3x^5 + x^3 - x \].
Ensuring accurate polynomial manipulation lays the groundwork for advanced algebra.

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Most popular questions from this chapter

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