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Chapter 8: Problem 16

Find \((f+g)(x)\) and \((f+g)\) $$f(x)=4 x^{2}-x-3, g(x)=x+1$$

Short Answer

Expert verified
After the step-by-step solution, the values of \((f+g)(x)\) and \((f+g)\) are both found to be \( 4x^{2} + x - 2 \).

Step by step solution

01

Find (f+g)(x)

Add the function \(f(x)=4x^{2}-x-3\) and the function \(g(x)=x+1\) together. This involves adding the expressions of the two functions together component wise: \( (f+g)(x) = (4x^{2}-x-3) + (x+1)\).
02

Simplify the expression

By combining like terms, we can simplify the expression obtained in step 1. This involves first adding the \(x\) terms together: \(4x^{2} + 0x - 3 + x + 1 = 4x^{2} + x - 2\).
03

Check the resulting function

(f+g)(x) will be the expression obtained from the simplification in the previous step: \( (f+g)(x) = 4x^{2} + x - 2 \).
04

Find (f+g)

Since \( (f+g)(x) = 4x^{2} + x - 2, (f+g) \) is simply the function without the \(x\) variable: \( (f+g) = 4x^{2} + x - 2 \).

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Key Concepts

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

Combining Functions
To understand how to combine functions in algebra, it's essential to recognize that each function represents a relationship between variables. When we speak of combining functions, we generally mean performing operations with them just as we do with numbers. These operations can include addition, subtraction, multiplication, and division.
For example, if you have two functions, say f(x) and g(x), combining them by addition would involve creating a new function, often denoted as (f+g)(x). To find this new function, simply add the expressions for f(x) and g(x) together:
  • Identify the corresponding components from each function.
  • Add these components together.
  • Simplify the resulting expression, if possible.
In the exercise, we combined f(x) = 4x^2 - x - 3 and g(x) = x + 1 to get (f+g)(x) = (4x^2 - x - 3) + (x + 1). After adding, we will simplify to find the solution.
Simplifying Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form, making it easier to work with and understand. The process usually involves a few common steps:
  • Combining like terms, which are terms that have the same variable raised to the same power.
  • Applying the distributive property to eliminate parentheses.
  • Cancelling terms where possible.
To simplify effectively, it helps to organize terms neatly and to be thorough in combining them correctly. In our example, we simplified the expression (f+g)(x) = (4x^2 - x - 3) + (x + 1) by combining like terms. Here, -x and x are like terms, which when combined, give us 4x^2 + x - 2; this is our simplified expression for (f+g)(x).
Polynomial Arithmetic
Polynomial arithmetic refers to the operations you can perform on polynomials, which are expressions consisting of variables and coefficients, connected by addition, subtraction, multiplication, and sometimes division. When dealing with polynomials, keep in mind the following tips:
  • Always arrange the terms of the polynomials in descending order of their powers.
  • When adding or subtracting, line up the like terms to make sure you combine them correctly.
  • Be careful with signs while combining terms to avoid errors.
  • Multiplying polynomials requires the distributive property, often referred to as the FOIL method for binomials.
In our exercise, we performed polynomial addition, which is an essential part of polynomial arithmetic. To illustrate with a simple example, if we are adding (4x^2 - x - 3) and (x + 1), we are engaging in polynomial arithmetic to arrive at the combined function 4x^2 + x - 2.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The function \(f(x)=5\) is one-to-one.

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=3 x-1$$

Describe a procedure for finding \((f \circ g)(x)\).

\(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{c|c}\hline x & f(x) \\\\\hline-1 & 1 \\\\\hline 0 & 4 \\\\\hline 1 & 5 \\\\\hline 2 & -1 \\ \hline\end{array}$$ $$\begin{array}{c|c}\hline x & g(x) \\\\\hline-1 & 0 \\\\\hline 1 & 1 \\\\\hline 4 & 2 \\\\\hline 10 & -1 \\ \hline\end{array}$$ $$f(g(1))$$

Find a. \((f \circ g)(x)\), b. \((g \circ f)(x)\), c. \((f \circ g)(2)\). $$f(x)=5 x+2, \quad g(x)=3 x-4$$
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