Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\sqrt{x}+2, \quad g(x)=x^{2}+3\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Functions

First, let's understand the given functions. We have \(f(x) = \sqrt{x} + 2\) and \(g(x) = x^2 + 3\). We need to find the composite functions \(f(g(x))\) and \(g(f(x))\).

Step 2: Finding f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Starting with \(g(x) = x^2 + 3\), replace \(x\) in \(f(x) = \sqrt{x} + 2\) with \(x^2 + 3\). Thus, \(f(g(x)) = \sqrt{x^2 + 3} + 2\).

Step 3: Simplifying f(g(x))

Check if \(\sqrt{x^2 + 3}\) can be simplified or further evaluated. Since it is already simplified, the expression for \(f(g(x))\) is \(f(g(x)) = \sqrt{x^2 + 3} + 2\).

Step 4: Finding g(f(x))

To find \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Using \(f(x) = \sqrt{x} + 2\), substitute \(f(x)\) into \(g(x) = x^2 + 3\). The new expression becomes \(g(\sqrt{x} + 2) = (\sqrt{x} + 2)^2 + 3\).

Step 5: Simplifying g(f(x))

Simplify \(g(\sqrt{x} + 2) = (\sqrt{x} + 2)^2 + 3\). Expanding the square, we have \((\sqrt{x} + 2)^2 = (\sqrt{x})^2 + 2 \cdot 2 \cdot \sqrt{x} + 2^2 = x + 4\sqrt{x} + 4\). Adding 3 yields: \(x + 4\sqrt{x} + 7\). Hence, \(g(f(x)) = x + 4\sqrt{x} + 7\).

Key Concepts

Function Notation

Function notation is a compact and efficient way to describe how a function operates with its inputs to produce outputs. When you see something like \( f(x) \), it tells you that this function, named "\( f \)", has input \( x \). In our exercise, \( f(x) = \sqrt{x} + 2 \) and \( g(x) = x^2 + 3 \) tell us exactly how each function transforms input into output. Replace the \( x \) with any number from their respective domains to calculate a specific value.

• For \( f(x) = \sqrt{x} + 2 \), input \( x \) is first put under a square root, then 2 is added to it.
• For \( g(x) = x^2 + 3 \), you square the input \( x \) and then add 3 to it.

Function notation makes it easy to understand, communicate, and manipulate these mathematical operations, particularly in algebra where expressions can get complex.

Composition of Functions

The composition of functions involves taking two functions and combining them into one. This is done by plugging one function into another. In our exercise, we are working with \( f(g(x)) \) and \( g(f(x)) \).

What is Composition?

Composition is essentially a function being applied within another function. If you have a function \( f \) and another \( g \), then \( f(g(x)) \) means you first find \( g(x) \) and then use that result as the input for \( f \).

Finding \( f(g(x)) \)

- Start with \( g(x) \), which is \( x^2 + 3 \) in our case.- Substitute this into \( f(x) = \sqrt{x} + 2 \).- The result is \( f(g(x)) = \sqrt{x^2 + 3} + 2 \).

Finding \( g(f(x)) \)

- Begin with \( f(x) \), which is \( \sqrt{x} + 2 \).- Inject this into \( g(x) = x^2 + 3 \).- Simplify: \( g(f(x)) = (\sqrt{x} + 2)^2 + 3 = x + 4\sqrt{x} + 7 \).
This process can make complex transformations much clearer and more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a value. In this exercise, you'll see them represented in a few key ways.

• Expression of \( f(x) \): \( \sqrt{x} + 2 \) - Square root of \( x \) and an addition of 2.
• Expression of \( g(x) \): \( x^2 + 3 \) - \( x \) squared, plus a constant 3.
• The resulting expressions from compositions like \( f(g(x)) = \sqrt{x^2 + 3} + 2 \) and \( g(f(x)) = x + 4\sqrt{x} + 7 \) involve manipulating these terms.

Algebraic expressions form the building blocks in solving equations, understanding functions, and developing formulas. They need to adhere to mathematical rules and often require simplification to their most elementary forms.