Problem 44 - Algebra 2

Question

If \(f(x)=4 x, g(x)=2 x-1,\) and \(h(x)=x^{2}+1,\) find each value. $$ [h \circ(g \circ f)](2) $$

Step-by-Step Solution

Answer

The value is 226.

Step 1: Understand Composition of Functions

The notation \( h \circ (g \circ f) \) means we need to find \( h(g(f(x))) \). First, find \( g(f(x)) \) and then apply \( h(x) \) to the result.

Step 2: Evaluate Inner Function \( f(x) \)

Given that \( f(x) = 4x \), find \( f(2) \):\[ f(2) = 4 imes 2 = 8 \]

Step 3: Apply \( g(x) \) to Result of \( f(x) \)

Now use the result from Step 2 in \( g(x) = 2x - 1 \):\[ g(f(2)) = g(8) = 2 imes 8 - 1 = 16 - 1 = 15 \]

Step 4: Apply \( h(x) \) to Result of \( g(x) \)

Use the result from Step 3 in \( h(x) = x^2 + 1 \):\[ h(g(f(2))) = h(15) = 15^2 + 1 = 225 + 1 = 226 \]

Step 5: Write the Final Value

Thus, \( [h \circ (g \circ f)](2) = 226 \).

Key Concepts

Composition of Functions

In mathematics, the composition of functions involves combining two or more functions to create a new function. It is denoted by a small circle between the functions, such as \( h \circ g \), meaning "\( h \) of \( g \)." This process involves taking the output of one function and using it as the input for another function.
For example, given three functions \( f(x) = 4x \), \( g(x) = 2x - 1 \), and \( h(x) = x^2 + 1 \), the composition \( h \circ (g \circ f) \) consists of:

• First applying \( f \) to \( x \).
• Then applying \( g \) to the result of \( f \).
• Finally, applying \( h \) to the result of \( g \).

This process allows us to evaluate complex expressions by tackling them one step at a time.

Function Evaluation

Function evaluation refers to the process of finding the value of a function given a specific input. It involves substituting a number for the variable in the function's formula.
For example, to evaluate \( f(x) = 4x \) when \( x = 2 \), replace \( x \) with 2:

• \( f(2) = 4 \times 2 = 8 \)

This step-by-step approach is essential when dealing with composed functions, as it ensures the correct order of operations is followed. With a composition like \( h(g(f(x))) \), evaluating from the innermost function outward is critical.
By evaluating each function separately and in the correct order, we simplify complex chains of calculations and reduce the risk of errors.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to represent numbers, functions, and operations. It helps communicate mathematical ideas efficiently and accurately.
In function composition, notation plays an essential role. For instance:

• \( h \circ g \) indicates function composition, meaning "\( h \) of \( g \)."
• Square brackets \( [ ] \) are used to indicate that composition is evaluated first, as in \( [h \circ (g \circ f)](2) \).

Understanding this notation is key to correctly interpreting and solving function composition problems. It's vital to remember that order matters in function composition.
Using mathematical symbols and notation correctly allows for clear and organized solutions, making complex problems much easier to manage.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations, such as addition or multiplication, grouped together to perform calculations. In the context of functions, they represent the relationships between input and output values.
When composing functions like \( h(g(f(x))) \), each function \( f(x) = 4x \), \( g(x) = 2x - 1 \), and \( h(x) = x^2 + 1 \), are algebraic expressions.

• Each expression involves substituting the variable \( x \) with numbers or results from another function.
• Perform operations to simplify and solve these expressions step by step.

The step-by-step evaluation of these expressions helps manage complexity and reduce the chances of algebraic errors. Understanding how to manipulate algebraic expressions is crucial for success in mathematical problem solving.