Problem 53 - A Graphical Approach to Precalculus with Limits

Question

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=-10^{x}+4$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \log_{10}(4 - x) \).

Step 1: Replace f(x) with y

Start by replacing \( f(x) = -10^{x} + 4 \) with \( y = -10^{x} + 4 \). This will make it easier to work with the equation to find the inverse later.

Step 2: Swap x and y

To find the inverse of a function, swap the x and y variables. This means you now have the equation \( x = -10^{y} + 4 \).

Step 3: Isolate 10^{y}

To solve for \( y \), first isolate the term involving \( y \). Subtract 4 from both sides to get \( x - 4 = -10^{y} \).

Step 4: Remove the negative sign

Remove the negative sign on the right side by multiplying both sides by -1, yielding \( -(x - 4) = 10^{y} \) or \( 4 - x = 10^{y} \).

Step 5: Take the logarithm

Apply the logarithm to both sides to solve for \( y \). Recall that \( 10^{y} = z \) implies \( y = \log_{10}(z) \). Therefore, \( y = \log_{10}(4 - x) \).

Step 6: Write the Inverse Function

The inverse function \( f^{-1}(x) \) is \( \log_{10}(4 - x) \).

Step 7: Graphing Instructions

To graph the two functions, plot \( y = -10^{x} + 4 \) as \( Y_1 \) and \( y = \log_{10}(4 - x) \) as \( Y_2 \) in the same viewing window. These functions are reflections of each other across the line \( y = x \).

Key Concepts

Inverse Function

An inverse function essentially reverses the operations of the original function. For an exponential function like \( f(x) = -10^{x} + 4 \), finding its inverse allows us to determine what input \( x \) yields a particular output \( y \). In practical steps:

• We start by replacing \( f(x) \) with \( y \) for convenience, leading to \( y = -10^{x} + 4 \).
• We swap \( x \) and \( y \), resulting in \( x = -10^{y} + 4 \).
• Next, solve this equation for \( y \) to express the inverse: subtract \( 4 \), handle the negative, and apply a logarithm. This process unveils the inverse function \( f^{-1}(x) = \log_{10}(4 - x) \).

These steps illustrate how the operations in the original function are reversed in its inverse. Each transformation undoes the previous one.

Graphing

Graphing both \( f(x) \) and its inverse \( f^{-1}(x) \) on the same set of axes offers a visual representation of how these functions interact. To achieve this:

• Plot the original function \( y = -10^{x} + 4 \) first. This function typically exhibits exponential decay.
• Next, plot the inverse function \( y = \log_{10}(4 - x) \), a logarithmic curve.
• These graphs will appear as reflections across a common line known as \( y = x \). The crossing point at \( y = x \) represents where input and output are identical.

Graphing serves as a powerful tool to illustrate the symmetry inherent in functions and their inverses. It provides immediate visual cues about their relationship.

Logarithm

The logarithm function is a key tool for finding inverses of exponential functions. When dealing with an equation like \( 10^{y} = z \), we use logarithms because it provides a way to "undo" the exponential. In our solution:

• After isolating the exponential term, \( 10^{y} \), we take the base-10 logarithm of both sides.
• This transforms the equation into \( y = \log_{10}(z) \), making the inverse function possible.
• Logarithms capture the power to which the base must be raised to obtain a given number, which is essential in solving for \( y \) in exponentials like \( 10^{y} \).

Understanding how to apply logarithms effectively is crucial in calculus and higher-level mathematics, especially when working with inverses.

Function Reflection

Reflective symmetry in graphs is often observed between a function and its inverse. The line \( y = x \) acts as a mirror, showing how their inputs and outputs interchange. In this exercise:

• Plotting both \( y = -10^{x} + 4 \) and \( y = \log_{10}(4 - x) \) visually highlights their reflection over this diagonal line.
• This symmetry implies that if a point \((a, b)\) lies on \( f(x) \), then \((b, a)\) lies on \( f^{-1}(x) \).
• Understanding this reflection property aids in verifying if one function is indeed the inverse of another. If they reflect perfectly across \( y = x \), the inverse relationship holds.

Appreciating the concept of reflection in functions expands one's comprehension of symmetry and inverse functions beyond pure algebraic manipulation.