Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 47

Chapter 3, Problem 47

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=-2x+5

Verified step by step guidance

Start by identifying the given function: $f(x) = -2x + 5$.

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 $f(x+h) = -2(x+h) + 5$.

Next, find $f(x+h) - f(x)$ by subtracting the original function $f(x)$ from the expression you found for $f(x+h)$. This gives $f(x+h) - f(x) = [-2(x+h) + 5] - [-2x + 5]$.

Simplify the expression $f(x+h) - f(x)$ by distributing and combining like terms carefully.

Finally, to find $\frac{f(x+h) - f(x)}{h}$, divide the simplified expression from the previous step by $h$. This will give you the difference quotient, which is a fundamental concept in calculus and algebra for understanding rates of change.

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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 the 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 stepping stone toward concepts like average rate of change.

Difference Quotient

The difference quotient [ƒ(x+h) - ƒ(x)]/h represents the average rate of change of the function over the interval from x to x+h. It is a key concept in calculus, used to approximate derivatives, and helps understand the function's behavior as h approaches zero.

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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.

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