Problem 34 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=3-5 x\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = -\frac{1}{5}x + \frac{3}{5}\).

Step 1: Define the function equation

Start with the function given: \(f(x) = 3 - 5x\). First, let's write this equation in the form of \(y = f(x)\) for convenience. This gives us: \(y = 3 - 5x\).

Step 2: Swap the variables

To find the inverse, swap the roles of \(x\) and \(y\) in the equation. This results in: \(x = 3 - 5y\).

Step 3: Solve for the new 'y'

Rearrange the equation \(x = 3 - 5y\) to solve for \(y\): Subtract 3 from both sides: \(x - 3 = -5y\). Then, divide both sides by -5: \(y = \frac{x-3}{-5}\). Simplify to get \(y = -\frac{1}{5}x + \frac{3}{5}\).

Step 4: Write the inverse function

Now express \(y\) in terms of \(x\) as the inverse function: Thus, the inverse function is: \(f^{-1}(x) = -\frac{1}{5}x + \frac{3}{5}\).

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input for another. This is a key step in understanding inverse functions because it helps verify an essential property: an inverse function, denoted by \( f^{-1}(x) \), when composed with the original function \( f(x) \), results in the identity function. This means that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

To break this down:

• Start with the function \( f(x) = 3 - 5x \).
• The inverse function derived was \( f^{-1}(x) = -\frac{1}{5}x + \frac{3}{5} \).
• When these functions are composed: \( f(f^{-1}(x)) = 3 - 5(-\frac{1}{5}x + \frac{3}{5}) = x \), confirming the composition results in the original input \( x \).

This verification step is crucial as it ensures that what was calculated as the inverse function actually undoes the effect of the original function, circling us back to our initial \( x \).

Linear Functions

Linear functions are functions of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. These functions have a constant rate of change and are graphically represented as straight lines in the coordinate plane. The slope \( a \) determines the steepness of the line, while \( b \) indicates the y-intercept where the line crosses the y-axis.

For the function \( f(x) = 3 - 5x \), it can be re-arranged into the standard linear form \( f(x) = -5x + 3 \), where \( a = -5 \) and \( b = 3 \):

• The slope \(-5\) signifies that for every one unit increase in \( x \), \( f(x) \) decreases by 5 units, highlighting a downward trend.
• The y-intercept \(3\) shows that the function passes through the point (0, 3) on the graph.

When finding the inverse, the linear nature of \( f \) means the inverse, \( f^{-1}(x) = -\frac{1}{5}x + \frac{3}{5} \), is also a linear function. This symmetry in transformations is a defining feature of linear functions.

Solving Equations

To find the inverse of a function, one of the critical steps involves solving equations. In this context, solving equations refers to isolating the variable of interest. Through systematic steps, each manipulation brings us closer to expressing the inverse relationship.

Here's how it applies in the example:

• Starting with \( x = 3 - 5y \) after interchanging \( x \) and \( y \).
• Subtract 3 from both sides to begin isolating \( y \): \( x - 3 = -5y \).
• Divide every term by \(-5\) to solve for \( y \): \( y = \frac{x-3}{-5} = -\frac{1}{5}x + \frac{3}{5} \).

Using these steps is crucial to ensure the inverse function is accurately determined. Solving equations is a fundamental skill in algebra that empowers students to handle transformations and changes of functions confidently. Each step serves a purpose in progressing towards a clear representation of the inverse function.