Problem 79 - Algebra 2

Question

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (f \circ g)(x)+h(x) $$

Step-by-Step Solution

Answer

The simplified function is \((f \circ g)(x) + h(x) = 2 x + 28 + |-2x+4|\).

Step 1: Compute (f ∘ g)(x)

\(f(g(x))\) means that we take the function \(f(x)=4 x\) and wherever we see an \(x\), we replace it with \(g(x)=\frac{1}{2} x+7\). So, \(f(g(x)) = 4 (\frac{1}{2} x+7) = 2 x + 28\).

Step 2: Calculate h(x)

The function \(h(x)=|-2x+4|\) is given to us, so there is no need to compute anything here.

Step 3: Compute (f ∘ g)(x) + h(x)

Now that we have calculated \(f(g(x)) = 2 x + 28\) and we already know \(h(x)=|-2x+4|\), we can add these two functions together to get \((f \circ g)(x) + h(x) = (2 x + 28) + |-2x+4|\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions made up of numbers, variables, and algebraic operations such as addition, subtraction, multiplication, division, and powers. In the given problem, you are working with three algebraic functions:

• \(f(x) = 4x\)
• \(g(x) = \frac{1}{2}x + 7\)
• \(h(x) = |-2x + 4|\)

The task involves composing these functions, which means substituting one function into another. In this case, you substitute \(g(x)\) into \(f(x)\), yielding the composition \(f(g(x))\). Simultaneously, you need to combine this composition with the function \(h(x)\) to achieve the final simplified expression.
Understanding algebraic functions is crucial because they provide a framework for analyzing and solving mathematical problems involving a sequence of operations on variables.
Function composition is a way to build new functions out of existing ones, which is particularly useful in complex equations where multiple relationships play a role.

Absolute Value Functions

Absolute value functions feature a special notation that affects their calculation and interpretation. The absolute value of a number is its distance from zero on the number line, ignoring any negative sign.
In the exercise, you encounter the absolute value function \(h(x) = |-2x + 4|\). This function indicates that regardless of the value of \(-2x + 4\), the output will always be non-negative. It helps to visualize this by considering both cases:

• If \(-2x + 4\) is positive or zero, then \(h(x) = -2x + 4\).
• If \(-2x + 4\) is negative, then \(h(x) = -(-2x + 4) = 2x - 4\).

The role of an absolute value function often includes ensuring non-negative outputs, essential for contexts where negative results may not make sense, like measuring distances or determining magnitudes.
In function combinations like in this exercise, handling absolute values correctly is crucial to accurately capture the behavior of the function in different input zones.

Function Simplification

Function simplification helps make expressions easier to work with, usually by reducing complexity or revealing more straightforward forms. When you simplify a composed function such as \((f \circ g)(x) + h(x)\), the objective is to find the most direct form of this expression.
Step by step, you work through:

• First, composing \(f(g(x))\) by replacing each instance of \(x\) in \(f(x) = 4x\) with \(g(x) = \frac{1}{2}x + 7\).
• Next, simplifying the result to get \(2x + 28\).
• Finally, adding the expression to \(h(x)\), which remains \(|-2x + 4|\).

Simplification often involves combining like terms, removing parentheses, and addressing any special function needs like absolute values. In this exercise, the simplification process results in a clean and concise expression: \((2x + 28) + |-2x + 4|\).
Breaking down complex composed functions into simpler elements helps highlight their essential behaviors, and is a foundational skill in algebra.