Problem 7 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? $$ y=4-3 x $$

Step-by-Step Solution

Answer

The inverse of the given function \(y=4-3x\) is \(y= \frac{{4-x}}{3}\), which is also a function.

Step 1: Swap x and y

To find the inverse of the function, swap the roles of \(x\) and \(y\). That is, replace \(y\) in the equation with \(x\) and \(x\) with \(y\). The swapped equation is: \(x=4-3y\).

Step 2: Solve for y

Rearrange the equation to solve for \(y\). First, move \(3y\) to the left-hand side and \(x\) to the right-hand side: \(3y= 4-x\). Then, divide by 3 to isolate \(y\): \(y= \frac{{4-x}}{3}\).

Step 3: Inverse Functionality

The vertical line test can be used to determine if the inverse is also a function. If any vertical line you draw cuts the graph at only one point, then the graph represents a function. In the case of \(y= \frac{{4-x}}{3}\), any vertical line will cross the graph at only one point, confirming that it is a function.

Key Concepts

Function Inverses

An inverse of a function essentially reverses the unique role of the input and output, meaning that if you have a function where the input is transformed to provide an output, the inverse function does the reverse. In simple terms, the inverse takes you back to where you started.

• To find an inverse, swap the positions of the dependent variable (usually represented as \(y\)) and the independent variable (\(x\)). This step reflects the reversal nature of an inverse function.
• The initial function is \(y = 4 - 3x\). By swapping \(x\) and \(y\), you receive \(x = 4 - 3y\), which illustrates the inverse relationship.

Once you have swapped \(x\) and \(y\), the next task is to solve the new equation for \(y\) to find the explicit expression of the inverse. This involves basic algebra and rearranging the equation to resolve \(y\).

Solving Equations

The process of solving equations involves algebraic manipulation to find the value of the variable, typically \(y\), in the equation. For inverses, this process helps express the inverse function explicitly.

• Take our swapped equation \(x = 4 - 3y\). To make \(y\) the subject of the formula, you'll move terms involving \(y\) to one side and isolate it.
• Start by rearranging to obtain \(3y = 4 - x\), in which you group \(y\) terms on one side.
• The last step is to divide through by 3 to separate \(y\): \(y = \frac{4 - x}{3}\).

This resulting expression \(y = \frac{4 - x}{3}\) is your inverse. It's a simplified process but essential to handle each step meticulously to avoid errors.

Vertical Line Test

To determine if this inverse is also a function, we can use the vertical line test. This test checks whether a function is valid by analyzing its graph.

• Draw various vertical lines across the graph of the obtained inverse \(y = \frac{4 - x}{3}\).
• Check if any vertical line, no matter where drawn, intersects the graph more than once.

If all the vertical lines cross the graph in just one spot each, the inverse functions as well.
In this scenario, the inverse passes the vertical line test, which confirms its functionality as a valid function. This is informative because it helps validate the function property of inversions efficiently.