Problem 59 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{x}-3 $$

Step-by-Step Solution

Answer

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

Step 1: Swap f(x) with y

To find the inverse of the function, first replace the function notation with y, so the equation becomes: \[ y = \frac{1}{x} - 3 \]

Step 2: Swap x and y

To find the inverse, interchange x and y in the equation. This represents the reversal of the roles of x and y: \[ x = \frac{1}{y} - 3 \]

Step 3: Solve for y

Now, solve this equation for y. Start by isolating \( \frac{1}{y} \):\[ x + 3 = \frac{1}{y} \]

Step 4: Take reciprocal

To isolate y, take the reciprocal of both sides of the equation:\[ y = \frac{1}{x+3} \]

Step 5: Rewrite as f inverse

Replace y with \( f^{-1}(x) \) to write the inverse function:\[ f^{-1}(x) = \frac{1}{x+3} \]

Key Concepts

Function Notation

Function notation is a way to represent functions concisely using symbols. It helps us understand how different values are mapped in a function. For instance, if we have a function \( f(x) = \frac{1}{x} - 3 \), the notation expresses the relationship between each input \( x \) and its corresponding output.

• \( f(x) \) is the function's name, and other information can often follow in parentheses.
• \( x \) is variable; it can change to demonstrate different underlying relationships.
• When finding the inverse function, the main idea is that you want to reverse this process—essentially undo the function to solve for the original input.

To express the inverse, we change the outputs and inputs, which means swapping \( x \) and \( y \). This swap indicates that for the inverse, an output value becomes the input and vice versa. Hence, the inverse function uses the notation \( f^{-1}(x) \), indicating we are looking for the input that gives a particular function output.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. When looking for an inverse function, we often need to solve the equation for \( y \) after swapping it with \( x \) in the function. This involves a series of systematic steps:

• Isolate the variable: Try to get \( y \) on one side by itself. This often requires performing mathematical operations to cancel out other terms or coefficients.
• Rearrange as needed: Use algebraic manipulations such as addition, subtraction, multiplication, or division to simplify the problem.
• Substitute back: Once the equation is solved for \( y \), substitute it back to express the inverse function.

In the example, after changing \( y \) and \( x \), isolating and rearranging leads to the equation \( y = \frac{1}{x+3} \). That solution is then represented as \( f^{-1}(x) \), showing the inverse.

Reciprocal

The reciprocal of a number is its multiplicative inverse, essentially flipping a fraction. Understanding the reciprocal is crucial when dealing with functions involving fractions, as it helps in isolating variables. If we take \( \frac{1}{y} \), its reciprocal will be \( y \).

• Reciprocal: \( \frac{1}{a} \) is the reciprocal of \( a \), assuming \( a eq 0 \).
• To solve an equation like \( x + 3 = \frac{1}{y} \), utilize the property of reciprocals to turn the equation into \( y = \frac{1}{x+3} \).

By using reciprocals, we simplify solving for \( y \), making the equation balanced and achieving the goal of expressing \( f^{-1}(x) \). It's a powerful tool in algebra for handling inverse problems effectively.