Problem 537 - Precalculus

Question

Find the inverse function \(f^{-1}\) for the exponential function \(f(x)=2 \cdot e^{x+1}-5\).

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \ln \left( \frac{x+5}{2} \right) - 1 \).

Step 1: Set up the Equation with y

Start by writing the function in terms of y instead of f(x). This sets the stage for solving for x. \[ y = 2 \cdot e^{x+1} - 5 \]

Step 2: Isolate the Exponential Expression

Add 5 to both sides of the equation to begin isolating the exponential term. \[ y + 5 = 2 \cdot e^{x+1} \]

Step 3: Solve for the Exponential Term

Divide both sides by 2 to further isolate the exponential expression. \[ \frac{y+5}{2} = e^{x+1} \]

Step 4: Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to solve for the exponent. \[ \ln \left( \frac{y+5}{2} \right) = x+1 \]

Step 5: Solve for x

Subtract 1 from both sides to isolate x. \[ x = \ln \left( \frac{y+5}{2} \right) - 1 \]

Step 6: Express the Inverse Function

Write the inverse function \(f^{-1}(x)\) by replacing y with x from the previous solution. \[ f^{-1}(x) = \ln \left( \frac{x+5}{2} \right) - 1 \]

Key Concepts

Exponential Function

An exponential function is a special kind of function in which a constant base is raised to a variable exponent. The most common exponential function is the natural exponential function, denoted as \( e^x \), where \( e \) is an irrational number approximately equal to 2.71828. These functions grow very rapidly, which makes them quite useful in modeling real-world phenomena such as population growth and radioactive decay.
Exponential functions have a consistent form:

• They are of the form \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants.
• The value \( e^{bx} \) represents the core exponential behavior.
• The constant \( a \) stretches or compresses the function vertically.

You will often encounter exponential functions in various fields of science and mathematics due to their unique properties. These functions have no x-intercepts and will always intersect the y-axis at \( y=a \) when \( x=0 \).
Understanding how to manipulate and apply these functions is crucial when solving equations involving exponential terms, such as finding their inverses.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the inverse of the natural exponential function \( e^x \). Essentially, applying the natural logarithm allows us to "undo" the exponential operation, which is very handy when solving equations involving exponential expressions.
The key properties of the natural logarithm include:

• \( \ln(e^x) = x \): This property shows that taking the natural logarithm of an exponential eliminates the exponential function.
• \( e^{\ln(x)} = x \): Similarly, raising \( e \) to the power of the natural logarithm of \( x \) returns \( x \).
• The natural logarithm is undefined for non-positive numbers, meaning it only accepts positive inputs.

When solving equations like the one in the exercise, applying the natural logarithm to both sides enables us to solve for the variable inside the exponential. This operation is crucial in determining the inverse of functions like the one given.

Function Composition

Function composition involves combining two functions such that the output of one function becomes the input of another. This is expressed as \( (g \circ f)(x) = g(f(x)) \). In the context of finding inverse functions, function composition plays an important role in verification.
Here's how it works:

• If \( f(x) \) and \( f^{-1}(x) \) are inverse functions, then \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This means applying the function and its inverse in succession returns the original input.
• Composition is used to check whether two functions are indeed inverses of each other.
• This concept ensures that the calculated inverse function accurately "undoes" the original function.

In the given problem, ensuring proper function composition helps confirm that \( f^{-1}(x) = \ln\left( \frac{x+5}{2} \right) - 1 \) is the correct inverse for \( f(x) = 2 \cdot e^{x+1} - 5 \). It's vital to understand how composition validates and connects functions, adding an extra layer of understanding to inverse calculations.