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.
$ƒ (x) = - 4 x + 2$

검증된 단계별 안내

Start by identifying the given function: $f(x) = -4x + 2$.

To find $f(x+h)$, substitute $x+h$ into the function in place of $x$. This means replacing every $x$ in the expression with $(x+h)$, so write $f(x+h) = -4(x+h) + 2$.

Next, find $f(x+h) - f(x)$ by subtracting the original function $f(x)$ from the expression you found for $f(x+h)$. This gives $f(x+h) - f(x) = [-4(x+h) + 2] - [-4x + 2]$.

Simplify the expression $f(x+h) - f(x)$ by distributing and combining like terms carefully. This will help you see the difference in terms of $h$.

Finally, to find $\frac{f(x+h) - f(x)}{h}$, divide the simplified expression from the previous step by $h$. This expression is important because it represents the average rate of change of the function over the interval from $x$ to $x+h$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 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.

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)=−4x+2ƒ(x)=-4x+2