Skip to main content
Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.3.43a
Chapter 1, Problem 1.3.43a

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.

Verified step by step guidance
1
Step 1: Recognize that the unit circle x^2 + y^2 = 1 can be split into four segments, each representing a one-to-one function. These segments correspond to the four quadrants of the circle.
Step 2: For each quadrant, determine the range of x-values (domain) and the corresponding y-values. The unit circle is symmetric, so consider the signs of x and y in each quadrant.
Step 3: In the first quadrant, both x and y are positive. The function f_1(x) can be expressed as y = sqrt(1 - x^2) with the domain 0 <= x <= 1.
Step 4: In the second quadrant, x is negative and y is positive. The function f_2(x) can be expressed as y = sqrt(1 - x^2) with the domain -1 <= x <= 0.
Step 5: In the third quadrant, both x and y are negative. The function f_3(x) can be expressed as y = -sqrt(1 - x^2) with the domain -1 <= x <= 0.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the unit circle equation x² + y² = 1, the domain must be determined by isolating y in terms of x, which leads to two distinct functions for the upper and lower halves of the circle. Understanding the domain is crucial for identifying valid inputs and ensuring the function behaves correctly within its defined limits.
Recommended video:
5:10
Finding the Domain and Range of a Graph

One-to-One Functions

A one-to-one function is a type of function where each output value is associated with exactly one input value, meaning no two different inputs produce the same output. In the context of the unit circle, splitting it into four one-to-one functions allows us to express the circle in a way that each function can be inverted. This property is essential for solving problems that require finding inverse functions or analyzing the behavior of the functions over their respective domains.
Recommended video:
05:50
One-Sided Limits

Implicit vs. Explicit Functions

Implicit functions are defined by an equation involving both variables, such as x² + y² = 1, while explicit functions express one variable directly in terms of another, like y = f(x). To find the formulas for the functions ƒ₁(x), ƒ₂(x), ƒ₃(x), and ƒ₄(x) from the unit circle, one must manipulate the implicit equation to derive explicit forms for y. This distinction is vital for understanding how to work with different types of functions in calculus.
Recommended video:
05:14
Finding The Implicit Derivative
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


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

Solving equations Solve each equation.


ln 3x + ln (x + 2) = 0

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)

1
views
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


If y= 3ˣ , then x = ³√y

2
views
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