Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=5 x^{2}+2, x \geq 0 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{\frac{x - 2}{5}} \), \( x \geq 2 \).

Step 1: Understanding the Function

The given function is \( f(x) = 5x^2 + 2 \) with the restriction \( x \geq 0 \). This restriction ensures that the function is one-to-one and its inverse will exist.

Step 2: Substitute and Swap

Start by replacing \( f(x) \) with \( y \) to get the equation \( y = 5x^2 + 2 \). Next, swap \( x \) and \( y \) to set up for finding the inverse: \( x = 5y^2 + 2 \).

Step 3: Solve for \( y \)

Isolate \( y \) by first subtracting 2 from both sides: \( x - 2 = 5y^2 \). Then divide both sides by 5: \( \frac{x - 2}{5} = y^2 \).

Step 4: Take the Square Root

Take the square root of both sides to solve for \( y \). Since \( x \geq 0 \) and \( \frac{x - 2}{5} \geq 0 \), we only consider the positive square root: \( y = \sqrt{\frac{x - 2}{5}} \).

Step 5: Re-express as Inverse Function

The inverse function is expressed as \( f^{-1}(x) = \sqrt{\frac{x - 2}{5}} \). This represents the inverse of \( f(x) = 5x^2 + 2 \) with \( x \geq 0 \). Verify by substituting back to ensure \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Key Concepts

One-to-One Function

A one-to-one function is a special type of function where each output value corresponds to exactly one input value. This unique property means that no two different inputs point to the same output. Therefore, for a function to have an inverse, it must be one-to-one. To determine if a function is one-to-one, you can use the Horizontal Line Test.
If a horizontal line intersects the graph of the function at most once, then the function is one-to-one.
In our exercise, the function is given with a restriction, specifically ensuring that the function remains one-to-one. The restriction is that the domain of the function is only for values of \( x \geq 0 \).
This restriction helps in ensuring that the function values do not repeat, as squaring any negative numbers would otherwise give identical positive values, making the function non-one-to-one.

Function Restriction

Function restriction is an essential tool used to ensure that a function becomes one-to-one, thereby allowing an inverse to exist. In the given function \( f(x) = 5x^2 + 2 \), the restriction \( x \geq 0 \) plays a critical role.
This restriction modifies the domain of the function to only consider x-values that are zero or positive.
Why is this important? Because without this restriction, the squaring part of the function \( x^2 \) would produce non-unique outputs for positive and negative values.
Restricting the function's domain to \( x \geq 0 \) ensures that the function behaves as intended — every y-value (output) from the function corresponds to only one x-value (input). This careful restriction guarantees that the function is one-to-one.

Square Root

The square root operation is fundamental when it comes to finding the inverse of a function involving squared variables. When solving for an inverse, especially when dealing with quadratic functions, taking the square root becomes inevitable.
In our specific example, the function is \( f(x) = 5x^2 + 2 \).
After rearranging and simplifying, we ended up with \( y^2 = \frac{x - 2}{5} \).
To solve for \( y \), we take the square root of both sides: \( y = \sqrt{\frac{x - 2}{5}} \).
Notice that only the positive square root is considered. This is due to the restriction \( x \geq 0 \), ensuring positive values of \( y \). The square root sign here is not just solving an equation; it helps us in preserving the one-to-one nature by selecting the positive root.

Solving Equations

Solving equations systematically is a key part of finding the inverse of a function. When given a function like \( f(x) = 5x^2 + 2 \), the goal is to interchange the roles of x and y and solve for the new y. The steps to take are outlined clearly:

• Start by rewriting \( f(x) \) as \( y \).
• Swap the variables to get \( x = 5y^2 + 2 \).
• Isolate the term involving \( y^2 \). In this case, subtract 2 from both sides: \( x - 2 = 5y^2 \).
• Divide by 5 to simplify: \( \frac{x - 2}{5} = y^2 \).
• Finally, solve for \( y \) by taking the square root: \( y = \sqrt{\frac{x - 2}{5}} \).

Each step involves carefully manipulating the equation to isolate \( y \), bringing us to the final expression that represents the inverse of the original function. Ensuring each step is followed precisely will lead to the correct solution, reflecting the accurate inverse relation.