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Chapter 3: Problem 2

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

Short Answer

Expert verified
Question: Given the functions \(f(x) = x^{2}+2\) and \(g(x) = x^{2}-4x-2\), find \((f+g)(x)\), \((f-g)(x)\), and \((g-f)(x)\). Answer: \((f+g)(x) = 2x^2 - 4x, (f-g)(x) = 4x, (g-f)(x) = -4x-4\)

Step by step solution

01

Find \((f+g)(x)\)

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\) together. So, we will do this operation: $$(f+g)(x) = f(x) + g(x) = (x^{2}+2) + (x^{2}-4x-2).$$ Now we need to simplify the expression by combining like terms: $$(f+g)(x) = 2x^2 - 4x.$$
02

Find \((f-g)(x)\)

To find \((f-g)(x)\), we need to subtract the function \(g(x)\) from the function \(f(x)\). So, we will do this operation: $$(f-g)(x) = f(x) - g(x) = (x^{2}+2) - (x^{2}-4x-2).$$ Now we need to simplify the expression by combining like terms: $$(f-g)(x) = 4x.$$
03

Find \((g-f)(x)\)

To find \((g-f)(x)\), we need to subtract the function \(f(x)\) from the function \(g(x)\). So, we will do this operation: $$(g-f)(x) = g(x) - f(x) = (x^{2}-4x-2) - (x^{2}+2).$$ Now we need to simplify the expression by combining like terms: $$(g-f)(x) = -4x-4.$$ In summary, the three expressions are: $$(f+g)(x) = 2x^2 - 4x,$$ $$(f-g)(x) = 4x,$$ $$(g-f)(x) = -4x-4.$$

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Key Concepts

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

Function Addition
When we talk about function addition, we refer to the process where two functions are to be added together. Think of it as adding two simple numbers, but instead of numbers, we have functions with variables and possibly constants. In the context of our example, we add the functions f(x) and g(x) by adding the corresponding terms from each function. Here's how it can be done:

  1. Identify the like terms in both functions. In our case, the like terms are those with x^2, x, and the constants.
  2. Add the coefficients of the like terms together. For x^2, we have 1 (from f(x)) plus 1 (from g(x)) which equals 2, giving us 2x^2.
  3. Combine the constants in the same way. However, in our specific example, +2 and -2 cancel each other out.

So, the addition of f(x) and g(x) simplifies to 2x^2 - 4x. Easy, right? Just remember to always combine like terms to simplify the expression and make sure no terms are left uncombined.
Function Subtraction
Moving on to function subtraction, which might seem a tiny bit trickier than addition, but it works with the same basic principles. The only difference is that instead of adding coefficients, we subtract them. For the function (f-g)(x), here's what we do:

  1. As with addition, we identify the terms in each function that are alike.
  2. Then, subtract the coefficients of like terms: for the x^2 terms, we have 1 (from f(x)) minus 1 (from g(x)), which cancel each other out.
  3. Finally, subtract the constants in the same way. Here, the constants (+2 from f(x) and -2 from g(x)) combine to give us +4.

This results in the simplified function (f-g)(x) = 4x. The subtraction of g(x) from f(x) flips the sign of g(x)'s terms, which is crucial for getting to the right answer.
Simplifying Expressions
The art of simplifying expressions is essential in mathematics. This process involves reducing an expression to its most basic form without changing its value or meaning. For our functions, simplifying involves two main steps. First, we perform the operation (addition or subtraction), and then we tidy up by combining like terms. It's like cleaning your room: you might move things around, but ultimately, everything needs to be neatly in its place. This simplification helps to understand and work with the function more easily, and it allows for quicker identification of function properties like intercepts or symmetry.
Combining Like Terms
A key part of simplifying expressions is combining like terms. Like terms are terms within an expression that have the same variable raised to the same power. Here's how to combine them:

  • Look for terms that have exactly the same variable part, like x^2 or x.
  • Add or subtract their coefficients based on the operation you are performing.
  • Do not combine terms with different variable parts or powers; they are not like terms and remain separate in the expression.

For example, in our exercise, when we find (g-f)(x), we combine -4x from g(x) with the opposite of +4x from f(x) which then simplifies to -4. That's combining like terms in a nutshell. Keep an eye out for signs and make sure not to mix different variables or powers together.

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Most popular questions from this chapter

The table shows the average weekly earnings (including overtime) of production workers and nonsupervisory employees in industry (excluding agriculture) in selected years." $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\ \hline \text { Weekly Earnings } & \$ 191 & \$ 257 & \$ 319 & \$ 369 & \$ 454 & \$ 538 \\ \hline \end{array}$$(a) Use linear regression to find a function that models this data, with \(x=0\) corresponding to 1980 (b) According to your function, what is the average rate of change in earnings over any time period between 1980 and \(2005 ?\) (c) Use the data in the table to find the average rate of change in earnings from 1980 to 1990 and from \(2000-2005\) How do these rates compare with the ones given by the model? (d) If the model remains accurate, when will average weekly earnings reach \(\$ 600 ?\)

Determine the domain of the function according to the usual convention. $$f(t)=\sqrt{-t}$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=\sqrt{x^{3}+5} ; \quad g(x)=-\frac{1}{2} \sqrt{x^{3}+5}-6$$

The estimated number of 15 - to 24 -year-old people worldwide (in millions) who are living with HIV/AIDS in selected years is given in the table."$$\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2001 & 2003 & 2005 & 2007 & 2009 \\ \hline \begin{array}{l} \text { 15- to 24-year-olds } \\ \text { with HIV/AIDS } \end{array} & 12 & 14.5 & 17 & 19 & 20.5 \\ \hline \end{array}$$(a) Use linear regression to find a function that models this data, with \(x=0\) corresponding to 2000 (b) According to your function, what is the average rate of change in this HIV/AIDS population over any time period between 2001 and \(2009 ?\) (c) Use the data in the table to find the average rate of change in this HIV/AIDS population from 2001 to 2009 . How does this rate compare with the one given by the model? (d) If the model remains accurate, when will the number of people in this age group with HIV/AIDS reach 25 million?

Find the radius \(r\) and height \(h\) of a cylindrical can with a surface area of 60 square inches and the largest possible volume, as follows. (a) Write an equation for the volume \(V\) of the can in terms of \(r\) and \(h\). (b) Write an equation in \(r\) and \(h\) that expresses the fact that the surface area of the can is \(60 .\) [ Hint: Think of cutting the top and bottom off the can; then cut the side of the can lengthwise and roll it out flat; it's now a rectangle. The surface area is the area of the top and bottom plus the area of this rectangle. The length of the rectangle is the same as the circumference of the original can (why?).] (c) Write an equation that expresses \(V\) as a function of \(r\) [Hint: Solve the equation in part (b) for \(h\), and substitute the result in the equation of part (a).] (d) Graph the function in part (c), and find the value of \(r\) that produces the largest possible value of \(V\). What is \(h\) in this case?
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