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 down the given function: $f(x) = 1 - x^{2}$.

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

Next, calculate $f(x+h) - f(x)$ by subtracting the original function $f(x)$ from the expression you found for $f(x+h)$: $[1 - (x+h)^{2}] - [1 - x^{2}]$.

Simplify the expression $f(x+h) - f(x)$ by expanding $(x+h)^{2}$ and combining like terms carefully.

Finally, find $\frac{f(x+h) - f(x)}{h}$ by dividing the simplified difference by $h$, which sets up the difference quotient used in calculus.

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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 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 stepping stone toward concepts like average rate 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 key concept in calculus, representing the slope of the secant line, and is used to approximate derivatives as h approaches zero.

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