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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 52
Chapter 3, Problem 52

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)=−x2ƒ(x)=-x^2

Verified step by step guidance
1
Start by writing down the given function: \(f(x) = -x^2\).
To find \(f(x+h)\), substitute \(x+h\) into the function in place of \(x\). This means you calculate \(f(x+h) = - (x+h)^2\).
Next, find the difference \(f(x+h) - f(x)\) by subtracting the original function \(f(x) = -x^2\) from the expression you found for \(f(x+h)\).
Simplify the expression \(f(x+h) - f(x)\) by expanding the square \((x+h)^2\) and combining like terms carefully.
Finally, to find \(\frac{f(x+h) - f(x)}{h}\), divide the simplified difference by \(h\), and simplify the resulting 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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Function Composition

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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Product, Quotient, and Power Rules of Logs
Related Practice
Textbook Question

For each line, (a) find the slope and (b) sketch the graph. 2y = -3x

Textbook Question

Find the value of the function for the given value of x.

f(x)={3if 0<x≤410−2[[5−x]]if x>4, for , for x=6.2f(x)=\(\begin{cases}\)3 & \(\text{if }\)0<x\(\leq\)4\\ 10-2[\(\left\]\lbrack\)5-x]\(\right\[\rbrack\) & \(\text{if }\)x>4,\(\text{ for }\]\end{cases}\),\(\text{ for }\)x=6.2

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Textbook Question

For each line, (a) find the slope and (b) sketch the graph. y = 2x - 4

Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. ƒ(-3)

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Textbook Question

Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (1, 6), perpendicular to 3x+5y=1

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Textbook Question

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. y=√(x-3)