Problem 33 - Algebra Form and Function

Question

Find the inverse function. $$ p(x)=\frac{5-\sqrt{x}}{3+2 \sqrt{x}} $$

Step-by-Step Solution

Answer

The inverse function of the given function is: $$ p^{-1}(x) = \left(\frac{5 - 3x}{2x + 1}\right)^2 $$

Step 1: Replace p(x) with y

Rewrite the given function using y instead of p(x): $$ y = \frac{5-\sqrt{x}}{3+2 \sqrt{x}} $$

Step 2: Swap x and y in the equation

Replace x with y and y with x: $$ x = \frac{5-\sqrt{y}}{3+2 \sqrt{y}} $$

Step 3: Solve for y

Our goal now is to isolate y. We'll do this in several smaller steps. First, clear the denominators by multiplying both sides by $$3 + 2\sqrt{y}$$: $$ x (3 + 2\sqrt{y}) = 5 - \sqrt{y} $$ Next, distribute x on the left side: $$ 3x + 2x\sqrt{y} = 5 - \sqrt{y} $$ Now, move the term containing $$\sqrt{y}$$ to the left side and the constant term to the right side: $$ 2x\sqrt{y} + \sqrt{y} = 5 - 3x $$ Factorize $$\sqrt{y}$$ from the left side: $$ \sqrt{y}(2x + 1) = 5 - 3x $$ To isolate y, divide both sides by $$(2x + 1)$$: $$ \sqrt{y} = \frac{5 - 3x}{2x + 1} $$ Finally, square both sides to completely isolate y: $$ y = \left(\frac{5 - 3x}{2x + 1}\right)^2 $$

Step 4: Write the inverse function

Now that y is isolated, we can rewrite our function using the inverse notation: $$ p^{-1}(x) = \left(\frac{5 - 3x}{2x + 1}\right)^2 $$ So, the inverse function of the given function is: $$ p^{-1}(x) = \left(\frac{5 - 3x}{2x + 1}\right)^2 $$

Key Concepts

Algebraic manipulation

Algebraic manipulation is the process of using algebraic techniques to rearrange or simplify equations and expressions. In the context of finding an inverse function, it involves rearranging the original function to solve for the input variable in terms of the output variable. This can be done through a series of steps:

• Clearing fractions by multiplying both sides by a common denominator.
• Distributing terms to simplify expressions.
• Isolating terms that contain the variable of interest.
• Factoring to group similar terms together.
• Performing operations such as squaring or taking square roots to solve for the variable.

These steps were applied in the original solution to transform the given function into its inverse. The process involves carefully applying algebraic operations in the correct order. This ensures that both sides of the equation remain equal as you work towards isolating the variable.

Function notation

Function notation is a way to represent functions in mathematics using symbols for easy reference and manipulation. It allows us to express functional relationships between variables succinctly, using letters such as \( f(x) \), \( g(x) \), or sometimes \( p(x) \) as seen in this problem.

In the exercise, function notation was employed to manage the transformation from the original function to its inverse. The process began by expressing the original function \( p(x) = \frac{5 - \sqrt{x}}{3 + 2 \sqrt{x}} \).

Once the inverse was found, it was represented in inverse function notation as \( p^{-1}(x) = \left(\frac{5 - 3x}{2x + 1}\right)^2 \). This clearly distinguishes the inverse from the original function, using a compact notation that is easy to understand and work with.

Solving equations

Solving equations is a fundamental algebraic skill involving finding the value of unknown variables that make the equation true. In the case of inverse functions, it requires us to interchange the roles of the dependent and independent variable and solve for the new independent variable.

Starting from the rewritten function \( x = \frac{5 - \sqrt{y}}{3 + 2 \sqrt{y}} \), the solution steps involved isolating \( y \). This included:

• Multiplying both sides by \( 3 + 2\sqrt{y} \) to eliminate the denominator.
• Rearranging the resulting equation to collect all terms involving \( \sqrt{y} \) on one side.
• Factorizing and further simplifying to solve for \( \sqrt{y} \).
• Squaring both sides to fully isolate \( y \).

This process of solving equations helps in deducing the inverse function. It showcases how changing positions and solving methodically enable one to find relationships between variables in a function.