Problem 51 - 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. $$f(x)=4^{x}-3$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \log_4(x + 3) \).

Step 1: Understand the Given Function

The given function is \( f(x) = 4^x - 3 \). This is an exponential function where 4 is the base and \( x \) is the exponent. The function is shifted down by 3 units due to the \(-3\).

Step 2: Replace \( f(x) \) with \( y \)

Start by replacing \( f(x) \) with \( y \) to find its inverse. This gives us:\[ y = 4^x - 3 \]

Step 3: Solve for \( x \) in Terms of \( y \)

To find the inverse, solve for \( x \) in terms of \( y \). Add 3 to both sides:\[ y + 3 = 4^x \]Now, take the logarithm base 4 of both sides to solve for \( x \):\[ x = \log_4(y + 3) \]

Step 4: Express the Inverse Function

Now that we have \( x = \log_4(y + 3) \), express it in terms of \( y \) as the inverse function:\[ f^{-1}(y) = \log_4(y + 3) \]

Step 5: Graph \( f(x) \) and \( f^{-1}(x) \)

Graph \( f(x) = 4^x - 3 \) and its inverse \( f^{-1}(x) = \log_4(x + 3) \) on the same axis. \( f(x) \) will be an exponential curve approaching \( -3 \) from below as \( x \to -\infty \), and \( f^{-1}(x) \) will be a logarithmic curve, intersecting the y-axis at \( x = \log_4(3) \).

Key Concepts

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent. In this context, the given function is \( f(x) = 4^x - 3 \). Here, the base is 4, and the exponent is \( x \). This function represents an exponential growth that has been shifted downward by 3 units. This downward shift is due to the "-3" at the end.

• The base of the exponential function determines the rate of growth or decay. A base greater than 1, such as 4, results in growth.
• Subtracting at the end shifts the entire graph downward.

Exponential functions are widely used to model real-world phenomena like population growth and radioactive decay. Their distinct "J"-shaped curve signifies rapid growth or decay depending on the perspective.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. For the function \( f(x) = 4^x - 3 \), we found its inverse \( f^{-1}(x) = \log_4(x + 3) \). This inverse function allows us to find the exponent when the result of an exponential expression is known. Let's explore it in more detail:

• The logarithmic function, \( \log_4(x + 3) \), essentially tells us what power the base 4 must be raised to in order to produce \( x + 3 \).
• When finding the inverse of an exponential function, we solve for the exponent as a function of the result of the original exponential expression.

The conversion from exponential to logarithmic formats is vital in fields requiring complex equations, like information theory and acoustics, where understanding the growth rate is essential.

Graphing Functions

Graphing both functions on the same plane allows us to visually compare them. For \( f(x) = 4^x - 3 \), the graph is an exponential curve that moves upward sharply and approaches \(-3\) as \( x \) becomes more negative. Its inverse function, \( f^{-1}(x) = \log_4(x + 3) \), forms a logarithmic curve which gradually rises and intersects with the y-axis at the point where \( x = \log_4(3) \).

• Exponential graphs generally show rapid increases and are initially quite flat when \( x \) is negative.
• Logarithmic graphs, in contrast, start steep and flatten out as \( x \) increases.

When sketched together, these graphs demonstrate symmetry about the line \( y = x \), as exponential and logarithmic functions are perfect inverse pairs. Graphing effectively bridges abstract equations and tangible visual comprehension, beneficial for interpreting the fundamental connection between these functions.