Problem 72 - Algebra 2

Question

Find the inverse of each function, Is the inverse a function? $$ y=5-2 x $$

Step-by-Step Solution

Answer

The inverse of the function \(y = 5 - 2x\) is \(y = (5 - x) / 2\), and it is a function.

Step 1: Swap \(x\) and \(y\)

Start by replacing \(y\) with \(x\) and \(x\) with \(y\) to get \(x = 5 - 2y\)

Step 2: Solve for \(y\)

To find the inverse, isolate \(y\) on one side of the equation. This can be done by adding \(2y\) to both sides and then subtracting \(x\) from both sides to get \(2y = 5 - x\). Divide every term by \(2\) to solve for \(y\), resulting in \(y = (5 - x) / 2\). This is the inverse function.

Step 3: Test for function

To check if the inverse is a function, apply the horizontal line test. If any horizontal line drawn through the inverse function touches the function at more than one point, the inverse is not a function. For \(y = (5 - x) / 2\), any horizontal line drawn will intersect the function at exactly one point, therefore, \(y = (5 - x) / 2\) is a function.

Key Concepts

Solving Equations

To find the inverse of a given function, you need to solve an equation in a specific way. Begin by swapping the variables. In the case of the original function \( y = 5 - 2x \), you swap \( x \) and \( y \). This gives you \( x = 5 - 2y \).
After the swap, the goal is to isolate \( y \) on one side of the equation to solve for it. The key steps involve:

• Adding \( 2y \) to both sides to eliminate the negative sign on \( y \).
• Subtracting \( x \) from both sides to move \( x \) away from \( y \).
• Finally, dividing through by \( 2 \) to solve for \( y \).

By completing these steps, you'll solve for \( y \), thus finding the inverse function. In this case, the inverse is \( y = \frac{5-x}{2} \). Each of these steps are simple, but crucial in correctly determining the inverse.

Horizontal Line Test

The horizontal line test is a simple method used to determine if the inverse of a function is itself a function. This is an important concept because while every function may have an inverse, not all inverses are functions.
To perform the horizontal line test on an inverse function like \( y = \frac{5-x}{2} \):

• Draw horizontal lines across the graph of the function.
• Check if any horizontal line intersects the graph more than once.

If it does intersect more than once, the inverse is not a function. For the function \( y = \frac{5-x}{2} \), you will observe that any horizontal line will intersect the graph at exactly one point. This means the inverse is indeed a function.

Algebraic Manipulation

Algebraic manipulation is the technique used to rearrange and solve equations, particularly when finding the inverse of a function. In this process, you use arithmetic operations to transform equations to isolate variables.Here are some fundamental operations:

• Addition or subtraction to shift terms from one side of the equation to the other.
• Multiplication or division to simplify expressions and solve for a particular variable.

Consider our function \( x = 5 - 2y \). To isolate \( y \), first, you add \( 2y \) to both sides, resulting in \( 2y = 5 - x \).
Then, by dividing each term by \( 2 \), you manipulate the equation to \( y = \frac{5 - x}{2} \). This skill is essential not just for finding inverse functions, but also for solving a wide variety of other algebraic problems.