Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.11

Chapter 1, Problem 1.11

Sketch the graph of the inverse of ƒ. <IMAGE>

Verified step by step guidance

Step 1: Understand the concept of an inverse function. The inverse of a function \( f \), denoted as \( f^{-1} \), is a function that 'reverses' the effect of \( f \). For a function to have an inverse, it must be one-to-one (bijective).

Step 2: Identify the domain and range of the original function \( f \). The domain of \( f \) becomes the range of \( f^{-1} \), and the range of \( f \) becomes the domain of \( f^{-1} \).

Step 3: Reflect the graph of \( f \) across the line \( y = x \). This reflection will give you the graph of \( f^{-1} \). Each point \((a, b)\) on the graph of \( f \) corresponds to a point \((b, a)\) on the graph of \( f^{-1} \).

Step 4: Verify that the reflected graph is a function. Ensure that the graph of \( f^{-1} \) passes the vertical line test, which confirms that it is a valid function.

Step 5: Label the axes and key points on the graph of \( f^{-1} \) to clearly indicate the transformation from the original function \( f \).

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Key Concepts

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

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Graphing Functions and Their Inverses

When graphing a function and its inverse, a key property is that the graphs are reflections of each other across the line y = x. This means that if a point (a, b) lies on the graph of f, then the point (b, a) will lie on the graph of f⁻¹. Understanding this relationship is crucial for accurately sketching the inverse graph.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For inverse functions, the domain of the original function becomes the range of the inverse function and vice versa. Recognizing how to switch these sets is essential for correctly sketching the graph of an inverse function.

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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) = 2 / ( x² + 2)

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Textbook Question

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb x/y

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Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

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Textbook Question

Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

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Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.

cos⁻¹ (- 1/2 )