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)
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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.R.14
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))
Verified step by step guidance1
Understand that an odd function f satisfies the property f(-x) = -f(x) for all x in its domain.
Recognize that if f is one-to-one, it has an inverse function f⁻¹, and similarly for g.
To find f⁻¹(g⁻¹(4)), start by finding g⁻¹(4). This means finding the value x such that g(x) = 4.
Once you have the value of x from g⁻¹(4), use it to find f⁻¹(x). This means finding the value y such that f(y) = x.
Use the properties of odd functions and the given graphs to determine the specific values needed for f⁻¹ and g⁻¹.
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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 when analyzing the behavior of the function and its inverse, especially in relation to the values being computed.
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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 determining the existence of an inverse function, as only one-to-one functions can have well-defined inverses that map back uniquely to their original inputs.
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05:50One-Sided Limits
Inverse Functions
An inverse function essentially reverses the effect of the original function. If f is a function and f⁻¹ is its inverse, then f(f⁻¹(x)) = x for all x in the range of f. In this context, finding f⁻¹(g⁻¹(4)) involves first determining g⁻¹(4) and then applying f⁻¹ to that result, highlighting the importance of understanding how to compute and interpret inverse functions.
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4:49Inverse Cosine
Related Practice
Textbook Question
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
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>
f⁻¹ (10)
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(1 + f(-3))
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