Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 43

Chapter 3, Problem 43

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

Verified step by step guidance

Identify the given function: $f(x) = 2 - x$. This is a linear function where the output decreases as $x$ increases.

To find $f(x+h)$, substitute $x+h$ into the function in place of $x$. So, write $f(x+h) = 2 - (x + h)$.

Next, calculate $f(x+h) - f(x)$ by subtracting the original function $f(x) = 2 - x$ from the expression found in step 2: $[2 - (x + h)] - (2 - x)$.

Simplify the expression from step 3 by distributing the negative sign and combining like terms carefully.

Finally, find the difference quotient by dividing the result from step 4 by $h$: $\frac{f(x+h) - f(x)}{h}$. 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.

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, 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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