Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 45

Chapter 3, Problem 45

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=6x+2

Verified step by step guidance

Start by identifying the given function: $f(x) = 6x + 2$.

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 $f(x+h) = 6(x+h) + 2$.

Next, find $f(x+h) - f(x)$ by subtracting the original function $f(x) = 6x + 2$ from the expression you found for $f(x+h)$. This gives $f(x+h) - f(x) = [6(x+h) + 2] - (6x + 2)$.

Simplify the expression $f(x+h) - f(x)$ by distributing and combining like terms. This will help you see how the function changes when $x$ increases by $h$.

Finally, find the difference quotient by dividing the expression $f(x+h) - f(x)$ by $h$. Write this as $\frac{f(x+h) - f(x)}{h}$ and simplify if possible. This expression represents the average rate of change of the function over the interval from $x$ to $x+h$.

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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 step toward finding rates 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 foundational concept in calculus, representing the slope of the secant line, and is used to approximate derivatives.

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Related Practice

Textbook Question

Find the slope of the line satisfying the given conditions. through (5, 9) and (-2, 9)

Textbook Question

Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4. y=x²+5

Textbook Question

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.

$ƒ (x) = 4 x + 11$

Textbook Question

Determine whether each relation defines y as a function of x. Give the domain and range. y=√(4x+1)

Textbook Question

Find the value of the function for the given value of x. ƒ(x)=-[[-x]], for x=2.5

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Textbook Question

The graph of a linear function f is shown. (a) Identify the slope, y-intercept, and x-intercept. (b) Write an equation that defines f.