Problem 12 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x)) .\) Simplify your answers. $$f(x)=x^{2}+1, g(x)=\sqrt{x+2}$$

Step-by-Step Solution

Answer

\( f(g(x)) = x+3 \) and \( g(f(x)) = \sqrt{x^2 + 3} \).

Step 1: Identify Composite Functions

To find the composite function \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). Similarly, for \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \).

Step 2: Substitute and Simplify \( f(g(x)) \)

For \( f(g(x)) \), substitute \( g(x) = \sqrt{x+2} \) into \( f(x) = x^2 + 1 \):\[ f(g(x)) = (\sqrt{x+2})^2 + 1. \] Simplify this to get \( f(g(x)) = x + 2 + 1 = x + 3 \).

Step 3: Substitute and Simplify \( g(f(x)) \)

For \( g(f(x)) \), substitute \( f(x) = x^2 + 1 \) into \( g(x) = \sqrt{x+2} \):\[ g(f(x)) = \sqrt{(x^2 + 1)+2}. \] Simplify this to \( g(f(x)) = \sqrt{x^2 + 3} \).

Key Concepts

Understanding Function Composition

Function composition involves creating a new function by combining two existing functions. It is represented as \(f(g(x))\) or \(g(f(x))\). Essentially, this means replacing the \(x\) in one function with another function. When performing function compositions, it's important to follow the order of operations:

• First, evaluate the inner function.
• Then, plug the result into the outer function.

The concept is crucial because it allows us to understand more complex relationships between variables by creating a pathway through simpler functions. This puts a spotlight on the versatility and power of mathematics, as multiple layers of functions can lead to new expressions and insights.

Simplifying with Algebraic Manipulation

Algebraic manipulation is a method used to simplify expressions or equations. When dealing with function compositions, this skill becomes invaluable. The ultimate goal is to express a function in its simplest form, making it easier to understand and work with. For example:

• In the solution \(f(g(x)) = (\sqrt{x+2})^2 + 1\), we simplify by realizing that squaring the square root \((\sqrt{x+2})^2\) just gives us \(x+2\).
• Adding, we then have \(x+2+1 = x+3\).

This process of simplification not only makes our answers more concise but also exposes any underlying patterns. Mastering algebraic manipulation can enhance your ability to both compose functions and simplify resulting expressions.

Exploring the Square Root Function

The square root function, \(g(x) = \sqrt{x+2}\), is a type of radical function that outputs the principal square root of its input expression. Understanding its properties is key in function composition and beyond:

• It effectively "undoes" squaring, as seen when simplifying \((\sqrt{x+2})^2\), using the fact that \((\sqrt{a})^2 = a\).
• Square root functions naturally result in non-negative outputs for any given input \(x\) that satisfies the domain conditions, which in this case includes \(x \, \geq \, -2\).
• The domain of a square root function ensures that the values inside the square root are zero or positive, which is essential for having real number outputs.

Understanding these aspects of square root functions will help in solving complex equations and in the comprehension of transformations in calculus.