Textbook Question
Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)
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3. Functions
Function Operations
2:45 minutesProblem 15a
Textbook Question
Find the domain of each function. f(x) = 1/[4/(x - 1) - 2]
Verified step by step guidance1
Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the function is undefined when the denominator equals zero. Identify the denominator of the given function: \( \frac{3}{x - 1} - 2 \).
Step 2: Set the denominator equal to zero to find the x-values that make the function undefined: \( \frac{3}{x - 1} - 2 = 0 \).
Step 3: Solve the equation \( \frac{3}{x - 1} - 2 = 0 \). First, isolate the fraction by adding 2 to both sides: \( \frac{3}{x - 1} = 2 \).
Step 4: Eliminate the fraction by multiplying both sides of the equation by \( x - 1 \) (note: \( x - 1 \neq 0 \) to avoid division by zero): \( 3 = 2(x - 1) \). Expand and solve for x: \( 3 = 2x - 2 \), then \( 2x = 5 \), and finally \( x = \frac{5}{2} \).
Step 5: Combine the results. The function is undefined when \( x = 1 \) (from the denominator \( x - 1 \neq 0 \)) and when \( x = \frac{5}{2} \) (from solving \( \frac{3}{x - 1} - 2 = 0 \)). Therefore, the domain of the function is all real numbers except \( x = 1 \) and \( x = \frac{5}{2} \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that would make the denominator zero, as division by zero is undefined.
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Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 1/[3/(x - 1) - 2], the denominator must be analyzed to ensure it does not equal zero, which is crucial for determining the domain.
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Solving for Undefined Points
To find the domain, one must identify values of x that make the denominator zero. This involves setting the denominator equal to zero and solving for x. Any x-value that results in a zero denominator must be excluded from the domain.
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