For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=x2+3x+1

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 55

Chapter 3, Problem 55

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=x²+3x+1

Verified step by step guidance

Start by writing down the given function: \(f(x) = x^2 + 3x + 1\).

To find \(f(x+h)\), replace every \(x\) in the function with \((x+h)\), so write \(f(x+h) = (x+h)^2 + 3(x+h) + 1\).

Next, expand the expression for \(f(x+h)\) by applying the binomial expansion to \((x+h)^2\) and distributing the 3 over \((x+h)\).

Calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x)\) from the expanded \(f(x+h)\) expression. This will involve combining like terms.

Finally, find the difference quotient by dividing the expression \(f(x+h) - f(x)\) by \(h\), which gives \(\frac{f(x+h) - f(x)}{h}\). Simplify the result as much as possible.

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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 rates of change and forms the basis for concepts like average rate of change and derivatives.

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 the derivative as h approaches zero.

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