Textbook Question
Find the domain of each function. f(x) = 1/(x2+1) - 1/(x2-1)
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3. Functions
Function Composition
2:40 minutesProblem 19d
Textbook Question
For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-8
Verified step by step guidance1
First, recall that the function (f/g)(x) is defined as the quotient of f(x) and g(x), so write it as \( (f/g)(x) = \frac{f(x)}{g(x)} \).
Substitute the given functions into the quotient: \( (f/g)(x) = \frac{3x + 4}{2x - 8} \).
Next, determine the domain of f(x) which is \( f(x) = 3x + 4 \). Since this is a linear function, its domain is all real numbers, or \( (-\infty, \infty) \).
Determine the domain of g(x) which is \( g(x) = 2x - 8 \). This is also a linear function with domain \( (-\infty, \infty) \), but since g(x) is in the denominator of the quotient, we must exclude values where \( g(x) = 0 \).
Solve \( 2x - 8 = 0 \) to find the value to exclude from the domain of (f/g)(x). This gives \( x = 4 \). Therefore, the domain of (f/g)(x) is all real numbers except \( x = 4 \), or \( (-\infty, 4) \cup (4, \infty) \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Division (f/g)(x)
Dividing two functions f and g means creating a new function (f/g)(x) = f(x) / g(x). This involves substituting the expressions for f(x) and g(x) and simplifying the resulting fraction, ensuring the denominator is not zero.
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Multiplying & Dividing Functions
Domain of a Function
The domain of a function is the set of all input values (x) for which the function is defined. For (f/g)(x), the domain excludes values that make the denominator zero, as division by zero is undefined.
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Linear Functions
Both f(x) = 3x + 4 and g(x) = 2x - 8 are linear functions, meaning their graphs are straight lines. Understanding their form helps in simplifying (f/g)(x) and determining domain restrictions based on their zero points.
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