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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 1.11
Chapter 1, Problem 1.11

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

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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.
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Inverse Cosine

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.
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Inverse Tangent

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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Finding the Domain and Range of a Graph
Related Practice
Textbook Question

Use the graph of ƒ to find ƒ⁻¹ (2),ƒ⁻¹ (9), and ƒ⁻¹ (12) <IMAGE>

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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 f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).

f(x)=10−2xf\(\left\)(x\(\right\))=10^{-2x}

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Textbook Question

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.


log660\(\log\)_660

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Textbook Question

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

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Textbook Question

Evaluate each expression without a calculator.


a. log₁₀ 1000

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