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Chapter 2: Problem 24

Find the functions \(f+g, f-g, f g\), and \(\mathrm{f} / \mathrm{g}\) \(f(x)=x^{2}+1 ; g(x)=\sqrt{x+1}\)

Short Answer

Expert verified
The operations between the functions \(f(x) = x^2 + 1\) and \(g(x) = \sqrt{x+1}\) are as follows: \[(f+g)(x) = (x^2 + 1) + (\sqrt{x+1})\] \[(f-g)(x) = (x^2 + 1) - (\sqrt{x+1})\] \[(fg)(x) = (x^2 + 1)(\sqrt{x+1})\] \[\left(\frac{f}{g}\right)(x) = \frac{x^2 + 1}{\sqrt{x+1}}, \textrm{ except when } x=-1\]

Step by step solution

01

Find the sum of the functions, f+g

To find the sum of the functions, we simply add the expressions for the two functions: \[(f+g)(x) = f(x) + g(x) = (x^2 + 1) + (\sqrt{x+1})\]
02

Find the difference of the functions, f-g

To find the difference of the functions, we subtract the expression of g(x) from the expression of f(x): \[(f-g)(x) = f(x) - g(x) = (x^2 + 1) - (\sqrt{x+1})\]
03

Find the product of the functions, fg

To find the product of the functions, we multiply the expressions for the two functions: \[(fg)(x) = f(x) \cdot g(x) = (x^2 + 1)(\sqrt{x+1})\]
04

Find the quotient of the functions, f/g

To find the quotient of the functions, we divide the expression of f(x) by the expression of g(x). However, we should also mention the domain restrictions (i.e., the values of x for which the expression is undefined because of division by zero): \[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 1}{\sqrt{x+1}}\] The expression is undefined when \(g(x) = \sqrt{x+1} = 0\), which occurs when \(x = -1\). So, we should mention that the domain of the function \(\frac{f}{g}\) is all real numbers, except \(x = -1\). In conclusion, we have found the following expressions: \[(f+g)(x) = (x^2 + 1) + (\sqrt{x+1})\] \[(f-g)(x) = (x^2 + 1) - (\sqrt{x+1})\] \[(fg)(x) = (x^2 + 1)(\sqrt{x+1})\] \[\left(\frac{f}{g}\right)(x) = \frac{x^2 + 1}{\sqrt{x+1}}, \textrm{ except when } x=-1\]

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Key Concepts

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

Function Addition
Function addition is the process of summing the values of two or more functions. To add functions, you simply combine their expressions. For example, given two functions \(f(x) = x^2 + 1\) and \(g(x) = \sqrt{x+1}\), the addition is performed as follows:

\[(f+g)(x) = f(x) + g(x) = (x^2 + 1) + (\sqrt{x+1})\].

This operation is straightforward. Just sum up corresponding parts from each function. If more complex terms are involved, ensure aligning like terms to simplify expressions quickly. This step doesn't change the domain of the resulting function unless specified by the operations inside each original component.
Function Subtraction
Function subtraction involves taking the difference between the values of two functions. It is much like function addition but instead, you subtract the expressions. In our example:

\[(f-g)(x) = f(x) - g(x) = (x^2 + 1) - (\sqrt{x+1})\].

When subtracting functions, be careful with signs and rearranging. Subtraction impacts the resulting function, especially when dealing with variable terms. After subtraction, check the domain limitations from each function. Any restrictions should carry over to prevent invalid results.
Function Multiplication
Function multiplication means multiplying two function expressions together. This combines the two expressions into a product, which might increase the complexity of manipulation. For example, for functions \(f(x) = x^2 + 1\) and \(g(x) = \sqrt{x+1}\):

\[(fg)(x) = f(x) \cdot g(x) = (x^2 + 1)(\sqrt{x+1})\].

After multiplication, simplify the expression if possible, and analyze for any domain restrictions that arise from the operation. Multiplication often broadens the scope of possible values, but it can also inherit the domain restrictions from the individual functions.
Function Division
Function division involves dividing one functional expression by another. This requires extra caution, particularly regarding division by zero issues. Consider \(f(x) = x^2 + 1\) and \(g(x) = \sqrt{x+1}\):

\[ \left(\frac{f}{g}\right)(x) = \frac{x^2 + 1}{\sqrt{x+1}}\].

This operation demands checking when the denominator becomes zero, as it introduces undefined points in our domain. In this example, the function \(g(x) = \sqrt{x+1} = 0\) when \(x = -1\). Thus, the division results in a function with a domain of all real numbers except \(x = -1\).

Ensure dividing without computational mistakes and consider any restrictions affecting possible function values. Division can greatly alter expected results if domain limits are overlooked.

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Most popular questions from this chapter

BREAK-EvEN ANALYSIS AutoTime, a manufacturer of 24 -hr variable timers, has a monthly fixed cost of \(\$ 48,000\) and a production cost of \(\$ 8\) for each timer manufactured. The units sell for \(\$ 14\) each. a. Sketch the graphs of the cost function and the revenue function and thereby find the break-even point graphically. b. Find the break-even point algebraically. c. Sketch the graph of the profit function. d. At what point does the graph of the profit function cross the \(x\) -axis? Interpret your result.

MARKET EQUILIBRIUM The management of the Titan Tire Company has determined that the weekly demand and supply functions for their Super Titan tires are given by $$ \begin{array}{l} p=144-x^{2} \\ p=48+\frac{1}{2} x^{2} \end{array} $$respectively, where \(p\) is measured in dollars and \(x\) is measured in units of a thousand. Find the equilibrium quantity and price.

BROADBAND INTERNET HousEHoLDS The number of U.S. broadband Internet households stood at 20 million at the beginning of 2002 and was projected to grow at the rate of \(6.5\) million households per year for the next 8 yr. a. Find a linear function \(f(t)\) giving the projected number of U.S. broadband Internet households (in millions) in year \(t\), where \(t=0\) corresponds to the beginning of \(2002 .\) b. What is the projected number of U.S. broadband Internet households at the beginning of 2010 ?

The percentage of obese children aged \(12-19\) in the United States is approximately \(P(t)=\left\\{\begin{array}{ll}0.04 t+4.6 & \text { if } 0 \leq t<10 \\\ -0.01005 t^{2}+0.945 t-3.4 & \text { if } 10 \leq t \leq 30\end{array}\right.\) where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1970 . What was the percentage of obese children aged \(12-19\) at the beginning of \(1970 ?\) At the beginning of 1985 ? At the beginning of 2000 ?

Patricia's neighbor, Juanita, also wishes to have a rectangular-shaped garden in her backyard. But Juanita wants her garden to have an area of \(250 \mathrm{ft}^{2}\). Letting \(x\) denote the width of the garden, find a function \(f\) in the variable \(x\) giving the length of the fencing required to construct the garden. What is the domain of the function? Hint: Refer to the figure for Exercise 26. The amount of fencing required is equal to the perimeter of the rectangle, which is twice the width plus twice the length of the rectangle.
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