Textbook Question
Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 2x² -3x +1
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.R.12
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>
f⁻¹ (10)
Verified step by step guidance1
Step 1: Understand the properties of odd functions. An odd function satisfies the condition f(-x) = -f(x) for all x in the domain of f.
Step 2: Recognize that f is one-to-one, meaning that each y-value in the range of f corresponds to exactly one x-value in the domain of f.
Step 3: Recall that the inverse function f⁻¹(y) gives the x-value such that f(x) = y.
Step 4: Since f is odd, if f(a) = 10, then f(-a) = -10. Use this property to find the x-value that corresponds to f(x) = 10.
Step 5: Use the graph of f to locate the x-value where f(x) = 10. This x-value is f⁻¹(10).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Odd Functions
An odd function is defined by the property f(-x) = -f(x) for all x in its domain. This symmetry about the origin implies that the graph of an odd function will reflect across both axes. Understanding this property is crucial for analyzing the behavior of the function and its inverse, especially when determining specific function values.
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Properties of Functions
One-to-One Functions
A one-to-one function, or injective function, is one where each output is produced by exactly one input. This means that if f(a) = f(b), then a must equal b. This property is essential for the existence of an inverse function, as it ensures that the inverse will also be a function, allowing us to find values like f⁻¹(10) without ambiguity.
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05:50One-Sided Limits
Inverse Functions
An inverse function essentially reverses the effect of the original function. If f(x) gives an output y, then f⁻¹(y) will return the input x. For one-to-one functions, the inverse can be found by swapping the roles of x and y in the equation y = f(x). Understanding how to find and interpret inverse functions is key to solving problems involving function values like f⁻¹(10).
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Related Practice
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>
f⁻¹( g⁻¹(4))
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) = 7 / (x + 3)
1
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Textbook Question
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
sin⁻¹ ( -1 )
6
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Textbook Question
Solving equations Solve each equation.
ln 3x + ln (x + 2) = 0
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