Question

The two functions share either an inside function or an outside function. Which is it? Describe the shared function. $$ y=\sqrt{5 x-2} \text { and } y=\sqrt{x^{2}+4} $$

Short answer

Answer: The shared function between the given functions is the outside function, which is the square root function, \(y=\sqrt{u}\).

Step-by-step solution

Identify the structure of the functions

Begin by understanding the overall structure of each function: For the first function: $$ y=\sqrt{5 x-2} $$ The structure can be identified as: \(y=\sqrt{u}\), where the inside function is \(u=5x-2\). For the second function: $$ y=\sqrt{x^{2}+4} $$ The structure can also be identified as: \(y=\sqrt{v}\), where the inside function is \(v=x^2+4\).

Compare the structures of both functions

Now that we know the structure of both functions, let's compare them to see if they share an inside or outside function: For the inside functions, \(u=5x-2\) and \(v=x^2+4\), we can see that they are not the same function. For the outside functions, in both cases, we have a square root function: \(y=\sqrt{u}\) and \(y=\sqrt{v}\). This means the outside function is shared, which is the square root function.

Describe the shared function

The shared function between these two given functions is the outside function, which is the square root function: $$ y=\sqrt{u} $$ In this case, \(u\) represents the inside function, which varies between the two given functions. The shared outside function is responsible for taking the square root of whatever is inside it.

Key concepts

Square Root Function

The square root function is a mathematical operation that takes the square root of a given number or expression. It is represented by the symbol \(\sqrt{}\) and can be expressed as \(y = \sqrt{x}\). In the functions provided, both make use of the square root function, where it acts as the outside function. This is a common mathematical approach to model phenomena that relate in a non-linear manner.

• The square root function only produces non-negative results when operated within the set of real numbers.
• The function is defined for all non-negative numbers, as the square root of a negative number is not defined within the real number system.
• It is also important in solving equations, especially in quadratic equations, to find unknown values when they are squared.

In our given examples, the square root function serves as the concluding operation, simplifying the final form of each function by determining the square root of the result of the inside functions (\(5x-2\) and \(x^2+4\)). Understanding how the square root function works helps explain how these functions curve and grow.

Inside Function and Outside Function

Inside and outside functions are key concepts in function composition. Essentially, they break down more complex functions into simpler parts—this is often useful for understanding how functions interact and transform pieces of data.

• The **inside function** is the first operation applied to the variable in the sequence of operations that forms the overall function. Example: \(u = 5x - 2\) or \(v = x^2 + 4\).
• The **outside function** operates on the result of the inside function. In our cases, it is the square root \(y = \sqrt{u}\) or \(y = \sqrt{v}\).

This layered approach allows mathematicians and learners to break down and analyze components of a function separately before understanding their interaction within a complete operation. It supports in simplifying complex expressions by dealing with smaller parts individually.

Comparing Functions

When comparing functions, the goal is to identify similarities and differences in their compositions and transformations. This can be assessed by analyzing their inside and outside functions.

• Identify if functions share the same structure or transform through investigation of common components like square root or polynomial forms.
• Check for shared outside functions like the square root, which was common in the two functions \(y=\sqrt{5x-2}\) and \(y=\sqrt{x^2+4}\).
• Notice differences in inside functions as they determine the specific transformations applied before the outside function takes effect.

In our example functions, both share the square root as an outside function, yet differ in their inside functionalities. Recognizing these distinctions helps in problem-solving as it allows one to predict how alterations in one section of the function can affect the overall outcome.