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Chapter 2: Problem 9

Find \(f+g, f-g, f g,\) and \(f / g\) and their domains. $$f(x)=x^{2}+x, \quad g(x)=x^{2}$$

Short Answer

Expert verified
(f+g)(x) = 2x^2 + x, (f-g)(x) = x, (f \cdot g)(x) = x^4 + x^3, (f/g)(x) = 1 + 1/x; domains: all reals (except f/g excludes 0).

Step by step solution

01

Find \( f+g \)

To find \( f+g \), we add the functions \( f(x) \) and \( g(x) \):\[f(x) + g(x) = (x^2 + x) + x^2 = 2x^2 + x\]The domain of \( f+g \) is the intersection of the domains of \( f \) and \( g \). Since both \( f(x) = x^2 + x \) and \( g(x) = x^2 \) are defined for all real numbers, the domain of \( f+g \) is all real numbers.
02

Find \( f-g \)

To find \( f-g \), we subtract \( g(x) \) from \( f(x) \):\[f(x) - g(x) = (x^2 + x) - x^2 = x\]The domain of \( f-g \) is also the intersection of the domains of \( f \) and \( g \), which is all real numbers.
03

Find \( f \cdot g \)

To find \( f \cdot g \), we multiply the functions \( f(x) \) and \( g(x) \):\[f(x) \, g(x) = (x^2 + x) \cdot x^2 = x^4 + x^3\]The domain of \( f \cdot g \) is all real numbers, as it is the product of polynomials, and they are defined for all real numbers.
04

Find \( \frac{f}{g} \)

To find \( \frac{f}{g} \), we divide \( f(x) \) by \( g(x) \):\[\frac{f(x)}{g(x)} = \frac{x^2 + x}{x^2}\]Simplifying gives:\[\frac{x^2 + x}{x^2} = 1 + \frac{x}{x^2} = 1 + \frac{1}{x}\]The domain of \( \frac{f}{g} \) requires that \( x^2 eq 0 \). Thus, \( x eq 0 \), and the domain is all real numbers except zero.

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Key Concepts

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

Function Operations
Function operations involve combining two functions using basic arithmetic processes: addition, subtraction, multiplication, and division. Each operation creates a new function.
  • **Addition (\( f+g \)):** This combines the outputs of \( f \) and \( g \) by adding them. For example, if \( f(x) = x^2 + x \) and \( g(x) = x^2 \), then \( f+g \) is calculated as \( f(x) + g(x) = 2x^2 + x \).
  • **Subtraction (\( f-g \)):** Here, you subtract the output of \( g \) from \( f \). Using the same functions, \( f-g \) is \( f(x) - g(x) = x \).
  • **Multiplication (\( f \cdot g \)):** This operation multiplies the outputs of the functions. For \( f \) and \( g \), \( f \cdot g \) becomes \( x^4 + x^3 \).
  • **Division (\( \frac{f}{g} \)):** This divides the output of \( f \) by \( g \). Be wary of division by zero; for our functions, \( \frac{f}{g} = 1 + \frac{1}{x} \).
Operations on functions can significantly alter their graphs and properties. Understanding how to perform these operations is crucial for exploring more complex mathematical relationships.
Domain of a Function
The domain of a function is the set of all possible input values (\( x \)) that the function can accept. Each fundamental operation can affect the domain differently.
  • **Addition and Subtraction:** When adding or subtracting functions, the domain of the result is typically the intersection of the domains of the individual functions. Given our example functions \( f(x) \) and \( g(x) \), both are polynomials with domain all real numbers, which makes \( f+g \) and \( f-g \) have the same domain: all real numbers.
  • **Multiplication:** The domain of the product \( f \cdot g \) is similarly the intersection of their domains. For polynomials, it's all real numbers.
  • **Division:** The domain of the quotient \( \frac{f}{g} \) excludes any \( x \) that makes \( g(x) = 0 \). In our case, \( x^2 \) being the denominator implies \( x eq 0 \), excluding zero from the domain.
Understanding the domain helps avoid undefined behavior and ensure the manipulation of these functions is valid over the entire set of inputs.
Polynomials
Polynomials are algebraic expressions made up of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. They are key components of many functions in algebra due to their simple properties and predictability over the real numbers.
  • **Structure:** A polynomial function \( p(x) \) is generally expressed as \( a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \), where \( a_n, a_{n-1}, ... a_0 \) are coefficients and \( n \) is a non-negative integer.
  • **Properties:** Polynomials are continuous and smooth, meaning they do not have breaks or bends, making their graphs easy to predict and analyze. This characteristic holds true irrespective of the degree of the polynomial.
  • **Domains:** Every polynomial has a domain of all real numbers, leading to their inclusion in the operations discussed. This universality simplifies the determination of domains for functions composed solely of polynomials.
Polynomials play a vital role in both theoretical and applied mathematics, forming the basis for many advanced mathematical concepts and real-world models.

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Most popular questions from this chapter

You place a frozen pie in an oven and bake it for an hour. Then you take the pie out and let it cool before eating it. Sketch a rough graph of the temperature of the pie as a function of time.

Graphing Functions Sketch a graph of the function by first making a table of values. $$f(x)=x^{2}-4$$

The table shows the number of DVD players sold in a small electronics store in the years 2003-2013. $$\begin{array}{|c|c|} \hline \text { Year } & \text { DVD players sold } \\ \hline 2003 & 495 \\ 2004 & 513 \\ 2005 & 410 \\ 2006 & 402 \\ 2007 & 520 \\ 2008 & 580 \\ 2009 & 631 \\ 2010 & 719 \\ 2011 & 624 \\ 2012 & 582 \\ 2013 & 635 \\ \hline \end{array}$$ (a) What was the average rate of change of sales between 2003 and 2013 ? (b) What was the average rate of change of sales between 2003 and 2004 ? (c) What was the average rate of change of sales between 2004 and 2005 ? (d) Between which two successive years did DVD player sales increase most quickly? Decrease most quickly?

DISCUSS: Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 7 we saw that the inverse of $$f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3}$$ because the "reverse" of "Multiply by 3 and subtract 2" is "Add 2 and divide by 3 ". Use the same procedure to find the inverse of the following functions. (a) \(f(x)=\frac{2 x+1}{5}\) (b) \(f(x)=3-\frac{1}{x}\) (c) \(f(x)=\sqrt{x^{3}+2}\) (d) \(f(x)=(2 x-5)^{3}\) Now consider another function: $$f(x)=x^{3}+2 x+6$$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(x)=\frac{1}{x^{n}}\) (a) \(n=1,3 ;[-3,3]\) by \([-3,3]\) (b) \(n=2,4 ;[-3,3]\) by \([-3,3]\) (c) How does the value of \(n\) affect the graph?
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