Textbook Question
For the pair of functions defined, find (ƒ-g)(x). Give the domain of each. See Example 2.
ƒ(x)=2x^2-3x, g(x)=x^2-x+3
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3. Functions
Function Composition
2:34 minutesProblem 23c
Textbook Question
For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2.
ƒ(x)=√(4x-1), g(x)=1/x
Verified step by step guidance1
Identify the composition of functions (ƒg)(x), which means ƒ(g(x)). This involves substituting g(x) into the function ƒ(x).
Write the expression for (ƒg)(x) by replacing x in ƒ(x) with g(x). Since ƒ(x) = \(\sqrt{4x - 1}\) and g(x) = \(\frac{1}{x}\), we have (ƒg)(x) = \(\sqrt{4 \cdot \frac{1}{x}\) - 1}.
Simplify the expression inside the square root: \(\sqrt{\frac{4}{x}\) - 1}.
Determine the domain of g(x) = \(\frac{1}{x}\). Since division by zero is undefined, the domain of g(x) is all real numbers except x = 0.
Determine the domain of (ƒg)(x) by considering the expression inside the square root must be greater than or equal to zero: \(\frac{4}{x}\) - 1 \(\geq\) 0. Solve this inequality along with the domain restriction from g(x) to find the domain of (ƒg)(x).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves applying one function to the result of another, denoted as (ƒg)(x) = ƒ(g(x)). It requires substituting the entire output of g(x) into the function ƒ. Understanding this process is essential to correctly combine the two given functions.
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Domain of a Function
The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of the composite function depends on the domains of both functions and the values for which the inner function's output fits the outer function's domain.
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Square Root and Rational Function Domains
For ƒ(x) = √(4x - 1), the expression inside the square root must be non-negative, so 4x - 1 ≥ 0. For g(x) = 1/x, x cannot be zero because division by zero is undefined. These restrictions are crucial when determining the domains of each function and their composition.
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