For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=6x+2

검증된 단계별 안내

Start by identifying the given function: \(f(x) = 6x + 2\).

To find \(f(x+h)\), substitute \(x+h\) into the function in place of \(x\). This means replacing every \(x\) in the function with \((x+h)\), so write \(f(x+h) = 6(x+h) + 2\).

Next, find \(f(x+h) - f(x)\) by subtracting the original function \(f(x) = 6x + 2\) from the expression you found for \(f(x+h)\). This gives \(f(x+h) - f(x) = [6(x+h) + 2] - (6x + 2)\).

Simplify the expression \(f(x+h) - f(x)\) by distributing and combining like terms. This will help you see how the function changes when \(x\) increases by \(h\).

Finally, find the difference quotient by dividing the expression \(f(x+h) - f(x)\) by \(h\). Write this as \(\frac{f(x+h) - f(x)}{h}\) and simplify if possible. This expression 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.