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Chapter 2: Problem 7

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

Short Answer

Expert verified
The functions are: \(f+g=3x\), \(f-g=-x\), \(fg=2x^2\), \(f/g=1/2\) for \(x \neq 0\). The domains for \(f+g, f-g, fg\) are all real numbers, and for \(f/g\) are all real numbers except 0.

Step by step solution

01

Add the Functions

To find \( f+g \), add the expressions for \( f(x) \) and \( g(x) \): \( f(x) + g(x) = x + 2x = 3x \). The domain is all real numbers since there is no restriction on \( x \) from both functions.
02

Subtract the Functions

To find \( f-g \), subtract the expression for \( g(x) \) from \( f(x) \): \( f(x) - g(x) = x - 2x = -x \). The domain is all real numbers, as there is no restriction on \( x \) from both functions.
03

Multiply the Functions

To find \( f \, g \), multiply the expressions for \( f(x) \) and \( g(x) \): \( f(x) \, g(x) = x \, (2x) = 2x^2 \). The domain is all real numbers since multiplication doesn't introduce any new restrictions.
04

Divide the Functions

To find \( f/g \), divide the expression for \( f(x) \) by \( g(x) \): \( \frac{f(x)}{g(x)} = \frac{x}{2x} = \frac{1}{2} \) for \( x eq 0 \). The domain excludes zero, as division by zero is undefined.

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Key Concepts

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

Addition of Functions
In mathematics, adding two functions is a straightforward process that involves adding their respective values for each input. For example, if we have the functions \( f(x) = x \) and \( g(x) = 2x \), their sum, \( f + g \), is found by simply combining them:
  • Calculate: \( f(x) + g(x) = x + 2x \)
  • Combine terms: \( 3x \)
This operation inherits its domain from both functions being added. Since both \( f(x) \) and \( g(x) \) allow all real numbers as input, the domain of \( f + g \) is also all real numbers. There are no restrictions in addition unless specified by the functions themselves.
Subtraction of Functions
Subtraction of functions follows a similar procedure as addition, but instead of adding, you are subtracting one function from the other. Given \( f(x) = x \) and \( g(x) = 2x \), the subtraction \( f - g \) is calculated by:
  • Subtract: \( f(x) - g(x) = x - 2x \)
  • Simplify: \( -x \)
The domain for subtraction, just like addition, will include all values where both original functions are defined. In this case, both functions are defined for all real numbers, so \( f - g \) has the domain of all real numbers.
Multiplication of Functions
When multiplying two functions, you simply multiply their outputs for given inputs. Let鈥檚 multiply the functions \( f(x) = x \) and \( g(x) = 2x \). Here's how to find \( fg \):
  • Perform multiplication: \( f(x) \cdot g(x) = x \cdot 2x \)
  • Simplify: \( 2x^2 \)
Similar to addition and subtraction, the domain for multiplying functions is usually the intersection of the domains of the original functions. Here, both \( f(x) \) and \( g(x) \) accept all real numbers, meaning \( fg \) also has a domain of all real numbers, void of any additional restrictions.
Division of Functions
Dividing functions is a bit more complex because division can introduce restrictions. To divide the functions \( f(x) = x \) and \( g(x) = 2x \), you must check where the second function is not zero:
  • Perform division: \( \frac{f(x)}{g(x)} = \frac{x}{2x} \)
  • Simplify: \( \frac{1}{2} \), as long as \( x eq 0 \)
Because division by zero is undefined, \( g(x) \) cannot be zero. This restriction affects the domain, making it all real numbers except zero. Always be aware of potential undefined conditions when dividing functions.

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