Textbook Question
Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure) <IMAGE>.
a. Find the domain and a formula for each function.
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 1.77a
Determine whether the following statements are true and give an explanation or counterexample.
If y= 3ˣ , then x = ³√y
Verified step by step guidance1
Consider the function \( y = 3^x \).
To solve for \( x \) in terms of \( y \), take the logarithm of both sides: \( \log(y) = \log(3^x) \).
Apply the logarithmic identity \( \log(a^b) = b \log(a) \) to get \( \log(y) = x \log(3) \).
Solve for \( x \) by dividing both sides by \( \log(3) \): \( x = \frac{\log(y)}{\log(3)} \).
Compare this expression with \( x = \sqrt[3]{y} \) to determine if they are equivalent.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form y = a^x, where 'a' is a positive constant and 'x' is the variable exponent. In this case, y = 3^x represents an exponential function where the base is 3. Understanding the properties of exponential functions is crucial for manipulating and solving equations involving them.
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Exponential Functions
Inverse Functions
An inverse function essentially reverses the effect of the original function. For an exponential function like y = 3^x, the inverse is found by solving for x in terms of y, leading to x = log₃(y). This concept is vital for determining relationships between variables and understanding how to express one variable in terms of another.
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Cube Root
The cube root of a number y, denoted as ³√y, is a value that, when multiplied by itself three times, gives y. This concept is important when analyzing the statement x = ³√y, as it implies a specific relationship between x and y that must be verified against the original exponential equation.
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12:57Summary of Curve Sketching Example 2
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(1 + f(-3))
Textbook Question
Inverse of composite functions
c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.
Textbook Question
Inverse of composite functions
b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹
Textbook Question
Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure)<IMAGE>.
b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)
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Textbook Question
Inverse of composite functions
a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹