For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=2-x

검증된 단계별 안내

Identify the given function: \(f(x) = 2 - x\). This is a linear function where the output decreases as \(x\) increases.

To find \(f(x+h)\), substitute \(x+h\) into the function in place of \(x\). So, write \(f(x+h) = 2 - (x + h)\).

Next, calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x) = 2 - x\) from the expression found in step 2: \([2 - (x + h)] - (2 - x)\).

Simplify the expression from step 3 by distributing the negative sign and combining like terms carefully.

Finally, find the difference quotient by dividing the result from step 4 by \(h\): \(\frac{f(x+h) - f(x)}{h}\). Simplify this expression as much as possible.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 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.