Problem 13 - College Algebra

Question

Use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. $$ f(x)=\sqrt{x}+2, g(x)=x^{2}+3 $$

Step-by-Step Solution

Answer

\(f(g(x)) = \sqrt{x^2 + 3} + 2; g(f(x)) = x + 4\sqrt{x} + 7\).

Step 1: Substitute g(x) into f(x)

Start by finding the composite function \(f(g(x))\). Substitute \(g(x) = x^2 + 3\) into \(f(x) = \sqrt{x} + 2\). This gives us \(f(g(x)) = \sqrt{x^2 + 3} + 2\).

Step 2: Simplify f(g(x) if possible

The expression \(\sqrt{x^2 + 3} + 2\) is already as simplified as possible given the functions provided. We do not know \(x\), so no further simplification can be done.

Step 3: Substitute f(x) into g(x)

Next, find the composite function \(g(f(x))\). Substitute \(f(x) = \sqrt{x} + 2\) into \(g(x) = x^2 + 3\). This results in \(g(f(x)) = (\sqrt{x} + 2)^2 + 3\).

Step 4: Expand and simplify g(f(x))

Expand \((\sqrt{x} + 2)^2\) to get \(x + 4\sqrt{x} + 4\). Adding 3, you get \(x + 4\sqrt{x} + 7\). Thus, \(g(f(x)) = x + 4\sqrt{x} + 7\).

Key Concepts

Function Operations

Understanding function operations is essential in calculus and algebra. When we talk about operations like addition, subtraction, multiplication, division, and composition of functions, we're essentially talking about how to combine functions in various ways. Function composition involves plugging one function into another. For example:

• If you have functions \( f(x) \) and \( g(x) \), the composite function \( f(g(x)) \) means "plug \( g(x) \) into \( f(x) \)."
• Alternatively, \( g(f(x)) \) means "plug \( f(x) \) into \( g(x) \)."

In our exercise, we composed the given functions and simplified the expressions. This demonstrates how composition combines functions to form a new function.

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It involves reducing an expression to its simplest form, making it easier to understand and work with. When simplifying, follow these steps:

• Combine like terms.
• Apply arithmetic operations correctly.
• Use algebraic identities when necessary, such as the square of a sum: \((a+b)^2 = a^2 + 2ab + b^2\).

For example, in \( g(f(x)) = (\sqrt{x} + 2)^2 + 3 \), we expanded the squared term and simplified the result, obtaining \( x + 4\sqrt{x} + 7 \). This process helps make expressions more manageable.

Square Root Functions

Square root functions are a type of radical function represented by \( \sqrt{x} \). These functions involve the operation of finding the square root of a number or expression. When dealing with square roots in function compositions, remember:

• The radicand (expression under the square root) must be non-negative since the square root of a negative number isn't a real number.

In this exercise,\( f(x) = \sqrt{x} + 2 \) is a square root function. We incorporate \( \sqrt{x} \) when we plug \( g(x) = x^2 + 3 \) into \( f(x) \), resulting in \( \sqrt{x^2 + 3} + 2 \). This highlights how square roots modify the domain of the function.

Polynomial Functions

Polynomial functions consist of sums of terms with non-negative integer exponents. The general form is \( a_n x^n + a_{n-1}x^{n-1} + ... + a_0 \). In our example, \( g(x) = x^2 + 3 \) is an elementary polynomial function.

• These functions are defined for all real numbers.
• The degree of the polynomial is the highest power of \( x \).

As seen in this exercise, when combining these with other functions like square root functions, each type of function can significantly alter the behavior and complexity of the resulting function. Understanding polynomials' basic structure helps simplify composite expressions such as \( g(f(x)) = (\sqrt{x} + 2)^2 + 3 \).