Problem 76 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified version of \((h \circ g)(x)\) is \(|x + 10|\)

Step 1: Understand function composition

The composite function \((h \circ g)(x)\) is obtained by substituting \(g(x)\) into \(h(x)\). This is written as \(h(g(x))\).

Step 2: Substitute \(g(x)\) into \(h(x)\)

So, to find \((h \circ g)(x)\), replace \(x\) in \(h(x)\) with \(g(x)\). This gives you \(h(g(x)) = |-2g(x) + 4|\). You know that \(g(x) = \frac{1}{2} x + 7\), so now replace \(g(x)\) with \(\frac{1}{2} x + 7\). This gives you the function \(h(g(x)) = |-2(\frac{1}{2} x + 7) + 4|\).

Step 3: Simplify the expression

Now, simplify the expression inside the absolute value function. This gives us \(|-x - 14 + 4|\), which further simplifies to \(|-x - 10|\). But this can be simplified further to \(|x + 10|\), as the absolute value does not change when multiplied by -1.

Key Concepts

Composite Function

Function composition is like nesting one function inside another. When you come across a composite function, such as \((h \circ g)(x)\), it means that you'll take the output of one function and use it as the input of another. In simpler terms, think of it as a two-step process:

• First execute the inside function, here it's \(g(x)\).
• Then, take the result and plug it into the outside function, which in this case is \(h(x)\).

So for \((h \circ g)(x)\), you perform \(h(g(x))\), leading to the next step in solving, which is substitution.

Absolute Value Function

An absolute value function expresses a number's distance from zero on a number line, disregarding direction. This means it always yields a non-negative result. The standard notation for absolute value is two vertical bars on either side of the expression, like so: \(|x|\). It tells us to consider only the magnitude of \(x\), ignoring whether it's positive or negative.
In our exercise, the function used is \(h(x) = |-2x + 4|\). Here, once the expression within the absolute value is simplified, any negative sign is ignored, transforming all results into positive magnitudes.

Simplifying Expressions

Simplifying expressions involves breaking down or reducing complex equations to a form that's easier to work with.

Why Simplify?

It's crucial because it makes equations more understandable and solvable. In this context, consider the expression inside the absolute value from our example: \|-2(\frac{1}{2} x + 7) + 4|.
The process includes:

• Distributing and combining like terms.
• Performing arithmetic operations, such as multiplication, division, and addition.

This results in \(-x - 14 + 4\), which further simplifies to \(-x - 10\). Given the absolute value, it turns into \(|x + 10|\), demonstrating the importance of simplification in solving the problem.

Substitution in Functions

Substitution is an essential technique where one function's result replaces the variable in another function. In this exercise, it involves inserting \(g(x) = \frac{1}{2}x + 7\) into \(h(x) = |-2x + 4|\).
By substituting, you change the variable in \(h(x)\) to represent the output of \(g(x)\). This transforms \(h(x)\) into \(h(g(x)) = |-2(\frac{1}{2}x + 7) + 4|\). You replace every occurrence of \(x\) in \(h(x)\) with \(g(x)\). This method helps evaluate complex compositions seamlessly by breaking them into manageable steps. It's like solving a puzzle, where each correct piece (or substitution) leads you towards the solution.