Textbook Question
In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)
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3. Functions
Intro to Functions & Their Graphs
1:21 minutesProblem 59
Textbook Question
Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. (fog) (0)
Verified step by step guidance1
Identify the functions involved: \(f(x) = 2x - 5\) and \(g(x) = 4x - 1\). The problem asks for \((f \circ g)(0)\), which means \(f(g(0))\).
First, evaluate the inner function \(g(0)\) by substituting \(x = 0\) into \(g(x) = 4x - 1\). This gives \(g(0) = 4(0) - 1\).
Simplify the expression for \(g(0)\) to find its value.
Next, take the result from \(g(0)\) and substitute it into the outer function \(f(x) = 2x - 5\). So, compute \(f(g(0)) = 2 \times g(0) - 5\).
Simplify the expression for \(f(g(0))\) to find the final value of \((f \circ g)(0)\).
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Video duration:
1mKey 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 (f ∘ g)(x) = f(g(x)). It means you first evaluate the inner function g at x, then use that output as the input for f. Understanding this process is essential to correctly evaluate composite functions without finding their explicit formula.
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Evaluating Functions at a Specific Input
Evaluating a function at a specific input means substituting the given value into the function's formula and simplifying. For composite functions, this requires careful step-by-step substitution, first evaluating the inner function at the input, then using that result in the outer function.
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Order of Operations in Function Evaluation
When evaluating composite functions, the order of operations is crucial: first compute the inner function's value, then apply the outer function to that result. This ensures accurate evaluation and avoids errors that arise from mixing up the sequence of substitutions.
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