Problem 9 - Brief Applied Calculus

Question

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt{x^{2}-9}+5 $$

Step-by-Step Solution

Answer

Let \(g(x) = x^2 - 9\) and \(f(x) = \sqrt{x} + 5\).

Step 1: Identify the Inner Function

To express the given function as a composition, we first identify the inner function. Notice that the expression inside the square root, \(x^2 - 9\), is a natural choice for the function \(g(x)\). Thus, let \(g(x) = x^2 - 9\).

Step 2: Define the Outer Function

Now that we have \(g(x) = x^2 - 9\), we consider the remaining part of the expression which involves \(\sqrt{g(x)} + 5\). This indicates that the outer function \(f(x)\) should be \(f(x) = \sqrt{x} + 5\).

Step 3: Compose the Functions

To check if the functions \(f\) and \(g\) we identified are correct, compose them: \(f(g(x)) = \sqrt{g(x)} + 5 = \sqrt{x^2 - 9} + 5\). This matches the original function, thus confirming our choices are correct.

Key Concepts

Inner Function

When working with function composition, an important concept is the inner function. In this exercise, the inner function is identified as the part of the expression that is inside another function's operation. For the given expression \( \sqrt{x^2 - 9} + 5 \), identifying the inner function involves looking at what lies inside the square root. Here, we have \( x^2 - 9 \) as the inner function. Therefore, we define it as \( g(x) = x^2 - 9 \).

Recognizing the inner function helps in breaking down complex expressions into manageable parts. This makes it easier to modify, evaluate, or graph these functions since each part can be tackled methodically.

Outer Function

The outer function is the function that operates on the result of another function. In our scenario, after identifying \( g(x) = x^2 - 9 \) as the inner function, the rest of the expression \( \sqrt{g(x)} + 5 \) implies that there is an external process occurring.

Thus, we see that \( f(x) = \sqrt{x} + 5 \) becomes the outer function that modifies the output of \( g(x) \). The outer function is crucial because it dictates how the result of the inner function is transformed or presented.

When analyzing or designing functions, understanding which operations occur at each level is vital for accurate computation or manipulation.

Square Root Function

The square root function, represented as \( \sqrt{x} \), is a vital mathematical concept found in numerous areas of math. It computes the non-negative root of a given input value. In the subject problem, the square root operation envelops the inner function \( x^2 - 9 \). This means that whatever result the inner function produces, the square root function will take that result and compute its square root.

This function has distinct properties, like being defined only for non-negative numbers (unless extended into the complex realm). It is continuous and increases slowly because, as values grow, their roots expand more sluggishly in comparison.

• When graphing, it produces a half-parabola-like curve.
• Helpful for rationalizing denominators in fractions.
• Used to determine geometric lengths, such as the hypotenuse in Pythagorean theorem.

Understanding its behavior, particularly in conjunction with specific transformations (like adding 5), can provide insights into how combined functions behave.