Question

Find the inverse of the function. \(y=e^{3 x}\)

Short answer

The inverse of the function \(y=e^{3 x}\) is \(y=\ln{x}/3\).

Step-by-step solution

Rewrite the function.

Rewrite the original function \(y=e^{3x}\) in terms of \(x\), we get \(x=e^{3y}\).

Take logarithm on both sides.

Take natural logarithm (\(\ln\)) on both sides to get rid of base \(e\), we obtain \(\ln{x}=\ln{e^{3y}} =3y\).

Solve for \(y\)

Solve the equation \(\ln{x}=3y\) for \(y\), we get \(y=\ln{x}/3\).

Key concepts

Exponential Functions

In understanding the concept of finding inverse functions, it is essential to grasp what exponential functions are. These functions are of the form f(x) = a^{x}, where the base a is a positive real number, and x is any real number. Exponential functions are known for their rapid growth or decay properties. This characteristic is extremely useful in modeling growth processes like population growth, radioactive decay, and interest compounding.

Characteristics of Exponential Functions

Exponential functions have some unique traits:

• Their output is always positive, as a positive number raised to any power remains positive.
• They grow at a rate proportional to their current value, which leads to an exponential increase or decrease.
• With a > 1, the function f(x) = a^{x} represents exponential growth. Conversely, with 0 < a < 1, it represents exponential decay.

It's these properties that we encounter when tackling problems involving finding the inverse of exponential functions, as we will apply algebraic manipulations and the properties of logarithms to rewrite the function in terms of the other variable.

Natural Logarithm

The natural logarithm, denoted as ln(x), is a special logarithm with base e where e is an irrational constant approximately equal to 2.71828. The natural logarithm serves as the inverse of the exponential function with base e; that is, ln(e^{x}) = x.

Using Natural Logarithm to Find Inverses

When dealing with exponential functions, taking the natural logarithm of both sides of an equation is a common technique used to solve for the exponent. Because the natural logarithm 'undoes' the exponent when the base is e, it simplifies the expressions and helps isolate the exponent.

For example, in solving for the inverse of y=e^{3x}, applying the natural logarithm to both sides removes the exponential portion, turning e^{3y} into 3y, thereby isolating the y term. This step is crucial in finding the inverse of the function, and it highlights the importance of understanding how natural logarithms work in relation to exponential functions.

Algebraic Manipulation

Algebraic manipulation involves the use of algebraic methods to rearrange and simplify equations. It's a fundamental skill in mathematics that paves the way for solving a variety of problems, including finding inverse functions. Mastery of algebraic manipulation techniques allows us to express a function in a different form, isolate variables, and solve equations.

Techniques in Algebraic Manipulation

Key techniques include:

• Using basic operations such as addition, subtraction, multiplication, and division to rearrange terms.
• Applying properties of exponents and logarithms to simplify expressions.
• Factoring and expanding polynomials when necessary.
• Isolating the variable of interest on one side of the equation.

Specific to finding the inverse of the function y = e^{3x}, algebraic manipulation covers the steps of taking the natural logarithm of both sides to eliminate the exponential term and then dividing by 3 to solve for y. This manipulation follows systematic and logical steps to transform an equation and ultimately find an inverse function that can describe a relationship in reverse.