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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 53
Chapter 3, Problem 53

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)=1−x2ƒ(x)=1-x^2

Verified step by step guidance
1
Start by writing down the given function: \(f(x) = 1 - x^{2}\).
To find \(f(x+h)\), substitute every \(x\) in the function with \((x+h)\), so write \(f(x+h) = 1 - (x+h)^{2}\).
Next, calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x)\) from the expression you found for \(f(x+h)\): \([1 - (x+h)^{2}] - [1 - x^{2}]\).
Simplify the expression \(f(x+h) - f(x)\) by expanding \((x+h)^{2}\) and combining like terms carefully.
Finally, find \(\frac{f(x+h) - f(x)}{h}\) by dividing the simplified difference by \(h\), which sets up the difference quotient used in calculus.

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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 as h approaches zero.
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Related Practice
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)=1+2x2ƒ(x)=1+2x^2

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Textbook Question

Find the slope of each line, provided that it has a slope. 11x + 2y = 3

Textbook Question

Graph each function. Give the domain and range. ƒ(x)=-[[x]]

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Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(10)

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Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(-2)

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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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