For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=1/x

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Start by writing the given function: \(f(x) = \frac{1}{x}\).

To find \(f(x+h)\), replace every \(x\) in the function with \((x+h)\), so \(f(x+h) = \frac{1}{x+h}\).

Next, calculate \(f(x+h) - f(x)\) by subtracting the original function from the new expression: \(\frac{1}{x+h} - \frac{1}{x}\).

To simplify \(f(x+h) - f(x)\), find a common denominator, which is \(x(x+h)\), and rewrite the expression as \(\frac{x - (x+h)}{x(x+h)}\).

Finally, to find \(\frac{f(x+h) - f(x)}{h}\), divide the simplified difference by \(h\), resulting in \(\frac{\frac{x - (x+h)}{x(x+h)}}{h}\), and simplify the complex fraction accordingly.

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Key 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 the output of a function for an input x. 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.

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 how the function varies over an interval and is a step toward finding rates of change.

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 foundational concept in calculus, representing the slope of the secant line, and is used to approximate derivatives.

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