For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.
$ƒ (x) = 1 / x^{2}$

Verified step by step guidance

Start by writing the given function: $f(x) = \frac{1}{x^2}$.

To find $f(x+h)$, replace every $x$ in the function with $(x+h)$, so write $f(x+h) = \frac{1}{(x+h)^2}$.

Next, calculate $f(x+h) - f(x)$ by subtracting the original function from the new expression: $\frac{1}{(x+h)^2} - \frac{1}{x^2}$.

To simplify $f(x+h) - f(x)$, find a common denominator, which is $x^2 (x+h)^2$, and rewrite the expression as a single fraction.

Finally, to find $\frac{f(x+h) - f(x)}{h}$, divide the simplified difference by $h$, which means multiplying the fraction by $\frac{1}{h}$.

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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 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 near x.

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 the difference quotient.

Difference Quotient and Its Role

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, used to approximate derivatives and analyze function behavior.

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