Textbook Question
Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.1.66
Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 2x² -3x +1
Verified step by step guidance1
Step 1: Start by substituting the function \( f(x) = 2x^2 - 3x + 1 \) into the difference quotient \( \frac{f(x+h) - f(x)}{h} \).
Step 2: Calculate \( f(x+h) \) by replacing \( x \) with \( x+h \) in the function: \( f(x+h) = 2(x+h)^2 - 3(x+h) + 1 \).
Step 3: Expand \( f(x+h) \): \( 2(x+h)^2 = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 \) and \( -3(x+h) = -3x - 3h \). Combine these to get \( f(x+h) = 2x^2 + 4xh + 2h^2 - 3x - 3h + 1 \).
Step 4: Substitute \( f(x+h) \) and \( f(x) \) into the difference quotient: \( \frac{(2x^2 + 4xh + 2h^2 - 3x - 3h + 1) - (2x^2 - 3x + 1)}{h} \).
Step 5: Simplify the expression by canceling out like terms and dividing each term by \( h \). This will give you the simplified form of the difference quotient.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Difference Quotient
The difference quotient is a formula used to find the average rate of change of a function over an interval. It is expressed as (ƒ(x+h) - ƒ(x))/h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the definition of the derivative, which measures the instantaneous rate of change.
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Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. In this case, we need to evaluate ƒ(x+h) and ƒ(x) for the given function ƒ(x) = 2x² - 3x + 1. Understanding how to correctly substitute and simplify expressions is crucial for manipulating the difference quotient.
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Algebraic Simplification
Algebraic simplification is the process of reducing expressions to their simplest form. This involves combining like terms, factoring, and canceling common factors. In the context of the difference quotient, simplifying the expression after substituting ƒ(x+h) and ƒ(x) is essential to arrive at a clear and concise result, especially as h approaches zero.
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Related Practice
Textbook Question
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sin⁻¹ ( -1 )
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Textbook Question
Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>
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Textbook Question
Solving equations Solve the following equations.
7ˣ = 21
Textbook Question
Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.
f(x) = x2 - 2x