Textbook Question
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
Function Composition
2:46 minutesProblem 55
Textbook Question
For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=x2+3x+1
Verified step by step guidance1
Start by writing down the given function: \(f(x) = x^2 + 3x + 1\).
To find \(f(x+h)\), replace every \(x\) in the function with \((x+h)\), so write \(f(x+h) = (x+h)^2 + 3(x+h) + 1\).
Next, expand the expression for \(f(x+h)\) by applying the binomial expansion to \((x+h)^2\) and distributing the 3 over \((x+h)\).
Calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x)\) from the expanded \(f(x+h)\) expression. This will involve combining like terms.
Finally, find the difference quotient by dividing the expression \(f(x+h) - f(x)\) by \(h\), which gives \(\frac{f(x+h) - f(x)}{h}\). Simplify the result as much as possible.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Notation and Evaluation
Function notation, such as ƒ(x), represents a rule that assigns each input x to an output. Evaluating ƒ(x+h) means substituting x+h into the function in place of x, which helps analyze how the function behaves when its input changes by h.
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Difference of Function Values
The expression ƒ(x+h) - ƒ(x) calculates the change in the function's output as the input changes from x to x+h. This difference is fundamental in understanding rates of change and forms the basis for concepts like average rate of change and derivatives.
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Difference Quotient
The difference quotient [ƒ(x+h) - ƒ(x)] / h measures the average rate of change of the function over the interval from x to x+h. It is a key concept in calculus, representing the slope of the secant line, and is used to approximate the derivative as h approaches zero.
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