Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{2}{x+8} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{2}{x} - 8\).

Step 1: Replace f(x) with y

Rewrite the function using y instead of f(x). This makes it easier to work with when finding the inverse: \[y = \frac{2}{x + 8}\]

Step 2: Switch x and y

To find the inverse, swap the variables x and y in the equation. This step sets up the equation for solving y in terms of x:\[x = \frac{2}{y + 8}\]

Step 3: Solve for y

Now solve the new equation for y. Start by multiplying both sides by \(y + 8\) to eliminate the fraction:\[x(y + 8) = 2\]Next, distribute x on the left side:\[xy + 8x = 2\]Isolate \(xy\):\[xy = 2 - 8x\]Finally, divide by x to solve for y:\[y = \frac{2 - 8x}{x}\]

Step 4: Simplify the expression

If possible, simplify the equation further. The expression can be rewritten as:\[y = \frac{2}{x} - 8\]

Step 5: Write the inverse function

The inverse function is formally written as follows:\[f^{-1}(x) = \frac{2}{x} - 8\]

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematics. It is commonly expressed as \( f(x) \), where \( f \) indicates the function name and \( x \) represents the input variable.
This notation helps in identifying what variable is being manipulated and how the output is determined.
In our exercise, the function is written as \( f(x) = \frac{2}{x+8} \). This tells us that when we input \( x \), the output is calculated by dividing 2 by the sum of \( x \) and 8.

• The "f" signifies a function, indicating to us that each input has a corresponding output.
• The "x" is the variable or input, which can change.
• The expression \( \frac{2}{x+8} \) is the rule or operation performed on the input \( x \).

These notations become particularly useful when we seek to find the inverse function as they help us see and understand the relationship between variable inputs and their outcomes.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to solve for variables or to change the form of the mathematical expression. In finding the inverse of a function, algebraic manipulation is crucial because it allows us to switch the roles of the dependent and independent variables.
In our given problem, the manipulation begins by swapping \( x \) and \( y \) in the equation \( y = \frac{2}{x+8} \) to form \( x = \frac{2}{y+8} \). This swap is pivotal as it forms the basis of finding the inverse. Following this, various steps are taken:

• Both sides of the equation are multiplied by \( y + 8 \) to eliminate fractions, giving us \( x(y + 8) = 2 \).
• The next step involves expanding by applying distributive law: \( xy + 8x = 2 \).
• We then isolate \( y \) by subtracting \( 8x \) from both sides: \( xy = 2 - 8x \).
• Dividing through by \( x \) gives \( y = \frac{2 - 8x}{x} \).

Each manipulation step advances us toward expressing the relationship where \( y \) is in terms of \( x \), revealing the inverse function.

Equation Solving

Solving equations is fundamental in mathematics when determining the value of unknowns. In solving for the inverse function, we need to express \( y \) solely in terms of \( x \). This task involves a series of calculated manipulations which help simplify the process.
Initially, we have the swapped equation \( x = \frac{2}{y+8} \). Solving it involves first eliminating the fraction:

• Multiply both sides by \( y + 8 \) resulting in: \( x(y + 8) = 2 \).
• Distribute \( x \) over \( y + 8 \) which translates to \( xy + 8x = 2 \).
• Rearrange to isolate terms involving \( y \): \( xy = 2 - 8x \).
• Finally, divide each term by \( x \): \( y = \frac{2 - 8x}{x} \).

To further simplify, the expression can be rewritten for clarity: \( y = \frac{2}{x} - 8 \). This process culminates in finding the inverse function that states \( f^{-1}(x) = \frac{2}{x} - 8 \), showing the input-output relationship reversal.