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.
$ƒ (x) = 4 x + 11$

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Start by finding $ f(x+h) $ by substituting $ x+h $ into the function $ f(x) = 4x + 11 $. This means replacing every $ x $ in the function with $ x+h $, so write $ f(x+h) = 4(x+h) + 11 $.

Next, simplify the expression for $ f(x+h) $ by distributing the 4 across $ (x+h) $, which gives $ 4x + 4h + 11 $.

Now, find $ f(x+h) - f(x) $ by subtracting the original function $ f(x) = 4x + 11 $ from the expression you found for $ f(x+h) $. Write this as $ (4x + 4h + 11) - (4x + 11) $.

Simplify the difference $ f(x+h) - f(x) $ by canceling out like terms. The $ 4x $ and $ 11 $ terms will cancel, leaving you with $ 4h $.

Finally, find the expression $ \frac{f(x+h) - f(x)}{h} $ by dividing the simplified difference $ 4h $ by $ h $. This will simplify to $ \frac{4h}{h} $, which you can simplify further by canceling $ h $ (assuming $ h \neq 0 $).

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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.

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 between two points on the function's graph.

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+11ƒ(x)=4x+11