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Chapter 0: Problem 53

\(f(x)=x^{2}+2 x+1, \quad g(x)=3 x+3\)

Short Answer

Expert verified
The sum of the functions is \(x^{2} + 5x + 4\), the difference is \(x^{2} - x - 2\), their product is \(3x^{3}+9x^{2}+12x+3\), and their quotient is \(\frac{x^{2}+2x+1}{3x+3}\).

Step by step solution

01

Find the Sum of the Functions

To find the sum of the functions, add \(f(x)\) and \(g(x)\) together. The sum is then \(h(x) = f(x) + g(x) = (x^{2}+2 x+1) + (3 x+3) = x^{2} + 5x + 4
02

Find the Difference of the Functions

To find the difference, subtract \(g(x)\) from \(f(x)\). So the difference is \(h(x) = f(x) - g(x) = (x^{2}+2 x+1) - (3 x+3) = x^{2} - x - 2
03

Find the Product of the Functions

To find the product, multiply \(f(x)\) by \(g(x)\). This gives \(h(x) = f(x) g(x) = (x^{2}+2 x+1) (3 x+3) = 3x^{3}+9x^{2}+12x+3
04

Find the Quotient of the Functions

Last but not least, to find the quotient, divided \(f(x)\) by \(g(x)\). This gives \(h(x) = \frac{f(x)}{g(x)} = \frac{x^{2}+2x+1}{3x+3}\)

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Key Concepts

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

Sum of Functions
In algebra, the sum of functions involves adding two functions together to form a new function. This operation is straightforward. Let's say you have two functions: \( f(x) = x^2 + 2x + 1 \) and \( g(x) = 3x + 3 \). The sum of these functions, denoted as \( h(x) = f(x) + g(x) \), is simply the result of adding every term in \( f(x) \) to every term in \( g(x) \). For our example, this results in:
  • Combining like terms: \( x^2 + 2x + 1 + 3x + 3 \)
  • Simplifying: \( x^2 + 5x + 4 \)
This new polynomial \( h(x) = x^2 + 5x + 4 \) is the sum of these two functions. It can be graphed or analyzed further, just like any other polynomial.
Difference of Functions
Finding the difference between two functions is another foundational concept in algebra. Here, you take one function and subtract the other. Using our previous functions, \( f(x) = x^2 + 2x + 1 \) and \( g(x) = 3x + 3 \), we achieve the difference through:
  • Subtraction operation: \( h(x) = f(x) - g(x) \)
  • Subtracting each term: \( x^2 + 2x + 1 - (3x + 3) \)
  • Simplifying: \( x^2 - x - 2 \)
The result \( h(x) = x^2 - x - 2 \) is a new function that represents the difference between \( f(x) \) and \( g(x) \). Each type of operation offers insight into how functions interact with one another.
Product of Functions
When finding the product of two functions, the goal is to multiply them together. This can create a more complex polynomial, reflecting the product of each paired term. Let's multiply \( f(x) = x^2 + 2x + 1 \) by \( g(x) = 3x + 3 \):
  • Start with distribution: Multiply each term of \( f(x) \) by each term of \( g(x) \)
  • Step-by-step multiplication gives: \( 3x^3 + 9x^2 + 12x + 3 \)
This result, \( h(x) = 3x^3 + 9x^2 + 12x + 3 \), illustrates how multiplying functions can increase the degree of the polynomial. It's a valuable operation for exploring the behavior of functions together.
Quotient of Functions
Dividing one function by another involves finding the quotient. This might yield a result that includes fractions or rational expressions. Using our functions, \( f(x) = x^2 + 2x + 1 \) and \( g(x) = 3x + 3 \), we find the quotient as follows:
  • Set up the division: \( h(x) = \frac{f(x)}{g(x)} \)
  • Which is \( \frac{x^2 + 2x + 1}{3x + 3} \)
Dividing functions can give insights into how the values of one function relate to another. Note that the quotient is only defined where \( g(x) eq 0 \), which is important in determining the domain of this resulting function.

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Most popular questions from this chapter

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ f^{-1} \circ g^{-1} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-2 x+15 & x_{1}=0, x_{2}=3 \end{array} $$

Prescription Drugs The amounts \(d\) (in billions of dollars) spent on prescription drugs in the United States from 1991 to 2002 (see figure) can be approximated by the model $$ d(t)= \begin{cases}5.0 t+37, & 1 \leq t \leq 7 \\ 18.7 t-64, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=1\) corresponding to 1991. Use this model to find the amount spent on prescription drugs in each year from 1991 to 2002 . (Source: U.S. Centers for Medicare \& Medicaid Services)

Path of a Ball The height \(y\) (in feet) of a baseball thrown by a child is $$ y=-\frac{1}{10} x^{2}+3 x+6 $$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

The total numbers \(f\) (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Federal Highway Administration) $$ \begin{array}{|c|c|} \hline 0 \text { Year, } t & \text { Miles traveled, } f(t) \\ \hline 5 & 2423 \\ 6 & 2486 \\ 7 & 2562 \\ 8 & 2632 \\ 9 & 2691 \\ 10 & 2747 \\ 11 & 2797 \\ 12 & 2856 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it mean in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}\) (2632). (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would \(f^{-1}\) exist? Explain.
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