Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The function \(f(x)=8^{x-3}-4\) is one-to-one. Find \(f^{-1}\).

Short answer

The inverse function is \[ f^{-1}(x) = \frac{\ln(x+4)}{\ln{8}} + 3 \].

Step-by-step solution

Understand the function

The given function is \( f(x) = 8^{x-3} - 4 \). To find the inverse function, \( f^{-1}(x) \), we need to follow specific steps involving algebraic manipulation.

Replace \( f(x) \) with \( y \)

Let's start by writing the function as an equation with \( y \): \[ y = 8^{x-3} - 4 \]

Solve for x

To find the inverse, solve for \( x \) in terms of \( y \). Add 4 to both sides: \[ y + 4 = 8^{x-3} \]

Apply the logarithm

Use the natural logarithm (ln) to solve for \( x \,\): \[ \ln(y+4) = \ln(8^{x-3}) \]

Simplify the equation

Use the property of logarithms \(\ln(a^b) = b\ln(a)\): \[ \ln(y+4) = (x-3)\ln(8) \]

Isolate \( x \)

Divide by \( \ln{8} \) and solve for \( x \): \[ x - 3 = \frac{\ln(y+4)}{\ln{8}} \]

Finalize the inverse function

Add 3 to both sides to solve for \( x \): \[ x = \frac{\ln(y+4)}{\ln{8}} + 3 \]

Rewrite inverse function

Now replace \( x \) with \( f^{-1}(x) \) and so \( y \) becomes \( x \): \[ f^{-1}(x) = \frac{\ln(x+4)}{\ln{8}} + 3 \]

Key concepts

one-to-one function

A one-to-one function is a type of function in which each value of the output (range) is paired with exactly one value of the input (domain). This means that no two different input values can map to the same output value. If a function is one-to-one, it guarantees that it has an inverse.

To check if a function is one-to-one, you can use the horizontal line test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one. This property ensures that solving for the inverse is possible.

In the example provided, the function \( f(x) = 8^{x-3} - 4 \) is one-to-one since the exponential function itself is one-to-one, and subtracting a constant (in this case, 4) does not change this property. Thus, we can proceed to find the inverse function \( f^{-1}(x) \).

logarithms

Logarithms are the inverse operations of exponentiation. If you have an equation of the form \( a = b^c \), the logarithm can help you solve for \( c \) by rewriting it as \( \log_b{a} = c \). This is useful when dealing with exponential functions.

In finding an inverse function for \( f(x) = 8^{x-3} - 4 \), one of the key steps is using logarithms to isolate the variable \( x \).

Let's look at how this works in our problem:

• Starting from \( y + 4 = 8^{x-3} \), we apply the natural logarithm \( \ln \) to both sides: \( \ln(y + 4) = \ln(8^{x-3}) \).
• Using the property \( \ln(a^b) = b \ln(a) \), this becomes \( \ln(y + 4) = (x-3) \ln(8) \).
• This logarithmic manipulation allows us to isolate \( x \), enabling the solving of the inverse function.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the desired variable. This is a fundamental skill in mathematics for solving equations and finding inverse functions.

For the function \( f(x) = 8^{x-3} - 4 \), we perform several steps of algebraic manipulation:

• First, replace \( f(x) \) with \( y \): \( y = 8^{x-3} - 4 \).
• To make the equation easier to work with, add 4 to both sides: \( y + 4 = 8^{x-3} \).
• Next, apply the natural logarithm: \( \ln(y + 4) = \ln(8^{x-3}) \).
• Use the property of logarithms to simplify: \( \ln(y + 4) = (x-3) \ln(8) \).
• Isolate the variable \( x \) by dividing both sides by \( \ln(8) \): \( x - 3 = \frac{\ln(y + 4)}{\ln(8)} \).
• Finally, add 3 to both sides to solve for \( x \): \( x = \frac{\ln(y + 4)}{\ln(8)} + 3 \).

These steps illustrate how algebraic manipulations help to unravel the function to find its inverse.

inverse operations

Inverse operations are mathematical processes that reverse the effects of another operation. For example, addition is the inverse of subtraction, and multiplication is the inverse of division.

When we talk about finding the inverse function, we are looking for a function that reverses the original function's effect. For the function \( f(x) = 8^{x-3} - 4 \), the inverse function \( f^{-1}(x) \) will output \( x \) when \( y \) (which is in the original function's range) is given.

To find the inverse, we replace \( f(x) \) with \( y \) and solve for \( x \) in terms of \( y \) using inverse operations:

• Add 4 to reverse the subtraction of 4: \( y + 4 = 8^{x-3} \).
• Apply logarithms to reverse the exponentiation: \( \ln(y + 4) = \ln(8^{x-3}) \).
• Simplify the logarithmic equation \( \ln(y + 4) = (x - 3) \ln(8) \) and solve for \( x \).
• Finally, translate back into \( y \): \( f^{-1}(x) = \frac{\ln(x+4)}{\ln(8)} + 3 \).

In summary, using inverse operations allows us to backtrack through the original function and reconstruct the input from the output.