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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 1.35
Chapter 1, Problem 1.35

Find the inverse f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).
f(x)=e2x+6f\(\left\)(x\(\right\))=e^{2x+6}

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Start by setting the function equal to y: y = e^{2x + 6}.
To find the inverse, swap x and y: x = e^{2y + 6}.
Solve for y by taking the natural logarithm of both sides: \(\ln\)(x) = 2y + 6.
Isolate y by first subtracting 6 from both sides: \(\ln\)(x) - 6 = 2y.
Finally, divide both sides by 2 to solve for y: y = \(\frac{\ln(x) - 6}{2}\). This is the inverse function, f^{-1}(x).

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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 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 back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input. This property ensures that the inverse function is well-defined.
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Exponential Functions

Exponential functions are of the form f(x) = a * e^(bx), where e is the base of natural logarithms, approximately equal to 2.71828. These functions grow rapidly and are characterized by their constant rate of growth proportional to their current value. Understanding their behavior is crucial for finding inverses, especially since the inverse of an exponential function is a logarithmic function.
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Exponential Functions

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are expressed as f^{-1}(x) = log_a(x) when the base is a. They help solve equations where the variable is an exponent. The properties of logarithms, such as the product, quotient, and power rules, are essential for manipulating and simplifying expressions when finding inverses of exponential functions.
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Related Practice
Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

Textbook Question

Solving equations Solve each equation.

√2 sin 3Θ + 1 = 2, 0 ≤ Θ ≤ π

Textbook Question

Finding inverses Find the inverse function.


ƒ(x) = 3x - 4

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

g(ƒ(u))

Textbook Question

Solving equations Solve the following equations.


ln x= -1

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Textbook Question

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


logbx²

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