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Chapter 2: Problem 10

Find \(f+g, f-g, f g,\) and \(f / g\) and their domains. $$f(x)=3-x^{2}, \quad g(x)=x^{2}-4$$

Short Answer

Expert verified
The results are \( f+g = -1 \), \( f-g = 7-2x^2 \), \( fg = -x^4 + 7x^2 - 12 \), and \( \frac{f}{g} = \frac{3-x^2}{x^2-4} \), with domains \( \mathbb{R} \), \( \mathbb{R} \), \( \mathbb{R} \), and \( \mathbb{R} \setminus \{-2, 2\} \), respectively.

Step by step solution

01

Addition of Functions

To find \( f + g \), we add the expressions for \( f(x) \) and \( g(x) \). \[ f(x) + g(x) = (3-x^2) + (x^2-4) \]Simplifying, we find:\[ f(x) + g(x) = 3 - x^2 + x^2 - 4 = -1 \]Therefore, \( f + g = -1 \) which is a constant function. The domain is all real numbers \( \mathbb{R} \).
02

Subtraction of Functions

To find \( f - g \), we subtract the expression for \( g(x) \) from \( f(x) \).\[ f(x) - g(x) = (3-x^2) - (x^2-4) \]Simplifying, we find:\[ f(x) - g(x) = 3 - x^2 - x^2 + 4 = 7 - 2x^2 \]Thus, \( f - g = 7 - 2x^2 \) and the domain is all real numbers \( \mathbb{R} \).
03

Multiplication of Functions

To find \( f \cdot g \), we multiply the expressions for \( f(x) \) and \( g(x) \).\[ f(x) \cdot g(x) = (3-x^2) \cdot (x^2-4) \]Expanding, we get:\[ f(x) \cdot g(x) = 3x^2 - 12 - x^4 + 4x^2 \]Combining like terms:\[ f(x) \cdot g(x) = -x^4 + 7x^2 - 12 \]The domain here is all real numbers \( \mathbb{R} \).
04

Division of Functions and Domain Constraints

To find \( \frac{f}{g} \), we divide \( f(x) \) by \( g(x) \).\[ \frac{f(x)}{g(x)} = \frac{3-x^2}{x^2-4} \]The domain of the division will exclude any \( x \) that makes the denominator zero. Setting \( x^2 - 4 = 0 \), we solve:\[ x^2 = 4 \]\[ x = \pm 2 \]Thus, the domain of \( \frac{f}{g} \) is all real numbers except \( x = 2 \) and \( x = -2 \). In interval notation, it is \( \mathbb{R} \setminus \{-2, 2\} \).

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Key Concepts

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

Domain of a Function
Understanding the domain of a function is crucial when working with mathematical operations. The domain refers to all possible input values (usually represented as "x") for which the function is defined. When considering individual functions like \( f(x) \, \text{and} \, g(x) \), it's important to note:
  • The domain of \( f(x) = 3 - x^2 \) is all real numbers \( \mathbb{R} \). This is because any real number "x" can be substituted into the equation without causing any issues, such as division by zero or taking the square root of a negative number.
  • Similarly, \( g(x) = x^2 - 4 \) also has a domain of all real numbers \( \mathbb{R} \), as you can compute \( x^2 - 4 \) for all real "x".

When combining functions, whether by addition, subtraction, multiplication, or division, checking the domain ensures that all work remains valid. For division, we must exclude values that cause the denominator to become zero, since division by zero is undefined.
Addition and Subtraction of Functions
When adding or subtracting functions, we simply combine like terms by performing the operations on their respective outputs:
  • For addition, \( f + g \), add the two functions together. For \( f(x) = 3 - x^2 \) and \( g(x) = x^2 - 4 \), we get:\[ f(x) + g(x) = (3-x^2) + (x^2-4) = -1 \]
  • The result is a constant value, and the domain for a constant function is always all real numbers \( \mathbb{R} \).
  • For subtraction, \( f - g \), subtract \( g(x) \) from \( f(x) \):\[ f(x) - g(x) = (3-x^2) - (x^2-4) = 7 - 2x^2 \]

This produces a quadratic expression with a domain of all real numbers since quadratic functions don't have restrictions unless they involve undefined operations such as division by zero.
Multiplication and Division of Functions
Multiplying and dividing functions involves combining their expressions, but special care is needed when considering their domains:
  • For multiplication, \( f \cdot g \), multiply the outputs of the functions:\[ f(x) \cdot g(x) = (3-x^2) \cdot (x^2-4) = -x^4 + 7x^2 - 12 \]
  • The result of the multiplication is a polynomial, which has a domain of all real numbers \( \mathbb{R} \).
  • Division, \( \frac{f}{g} \), needs caution as division by zero is undefined:\[ \frac{f(x)}{g(x)} = \frac{3-x^2}{x^2-4} \]
  • To determine the domain, set the denominator not equal to zero: \(x^2 - 4 = 0\) gives the solutions \( x = \pm 2 \).

Thus, the domain of \( \frac{f}{g} \) excludes \( -2 \) and \( 2 \), reflected as \( \mathbb{R} \setminus \{-2, 2\} \). Always check for these points in division to avoid errors in your calculations.

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Most popular questions from this chapter

Even and Odd Functions Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$f(x)=x+\frac{1}{x}$$

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Graphing Functions Sketch a graph of the function by first making a table of values. $$G(x)=|x|-x$$

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} 1-x & \text { if } x<-2 \\ 5 & \text { if } x \geq-2 \end{array}\right.$$

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(g(x)=x^{2}-x-20\) (a) \([-2,2]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-7,7]\) by \([-25,20]\) (d) \([-10,10]\) by \([-100,100]\)
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