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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 54
Chapter 3, Problem 54

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)=1+2x2ƒ(x)=1+2x^2

Verified step by step guidance
1
Start by writing down the given function: \(f(x) = 1 + 2x^2\).
To find \(f(x+h)\), substitute every \(x\) in the function with \((x+h)\), so write \(f(x+h) = 1 + 2(x+h)^2\).
Next, expand the squared term \((x+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\), which gives \(x^2 + 2xh + h^2\). Substitute this back into \(f(x+h)\).
Calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x) = 1 + 2x^2\) from the expression you found for \(f(x+h)\).
Finally, find the difference quotient by dividing the expression \(f(x+h) - f(x)\) by \(h\), which is \( rac{f(x+h) - f(x)}{h}\). Simplify this expression 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.
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Evaluating Composed Functions

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.
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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.
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