Textbook Question
Find the inverse of each function (on the given interval, if specified).
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 1.67
Intersection problems Find the following points of intersection.
The point(s) of intersection of the parabolas y= x² and y= -x² + 8x
Verified step by step guidance1
Step 1: Set the equations of the parabolas equal to each other to find the x-coordinates of the intersection points: \(x^2 = -x^2 + 8x\).
Step 2: Rearrange the equation to form a standard quadratic equation: \(2x^2 - 8x = 0\).
Step 3: Factor the quadratic equation: \(2x(x - 4) = 0\).
Step 4: Solve for x by setting each factor equal to zero: \(2x = 0\) and \(x - 4 = 0\).
Step 5: Substitute the x-values back into either original equation to find the corresponding y-values for the points of intersection.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Finding Points of Intersection
To find the points of intersection between two curves, we set their equations equal to each other. This involves solving the resulting equation for the variable, which will yield the x-coordinates of the intersection points. Once the x-values are found, they can be substituted back into either original equation to find the corresponding y-values.
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Critical Points
Quadratic Functions
Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax² + bx + c. The graphs of these functions are parabolas, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the properties of parabolas, such as their vertex, axis of symmetry, and direction of opening, is essential for analyzing their intersections.
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Solving Quadratic Equations
Solving quadratic equations can be done using various methods, including factoring, completing the square, or applying the quadratic formula. Each method has its advantages depending on the specific equation. For intersection problems, finding the roots of the resulting quadratic equation will reveal the x-coordinates of the intersection points, which are critical for determining where the two curves meet.
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Related Practice
Textbook Question
Finding all inverses Find all the inverses associated with the following functions, and state their domains.
ƒ(x) = (x + 1)³
Textbook Question
Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = - (3/√x)
Textbook Question
Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = √(1-2x)
Textbook Question
Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
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Textbook Question
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
cos (cos⁻¹ ( -1 ))
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