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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 46
Chapter 3, Problem 46

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

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1
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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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.
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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 step toward finding rates 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 foundational concept in calculus, representing the slope of the secant line between two points on the function's graph.
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Product, Quotient, and Power Rules of Logs
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