Problem 79 - Algebra 2

Question

Find \(g[h(x)]\) and \(h[g(x)]\) $$ \begin{array}{l}{h(x)=2 x+5} \\ {g(x)=-x+3}\end{array} $$

Step-by-Step Solution

Answer

\(g[h(x)] = -2x - 2\) and \(h[g(x)] = -2x + 11\)

Step 1: Understand Function Notation

The notation \(g[h(x)]\) represents the composite function where \(h(x)\) is evaluated first, and then \(g(x)\) is applied to the result of \(h(x)\). Similarly, \(h[g(x)]\) represents the composite function where \(g(x)\) is evaluated first, and then \(h(x)\) is applied to the result of \(g(x)\).

Step 2: Substitute for \(h(x)\) in \(g[h(x)]\)

Since \(h(x) = 2x + 5\), substitute \(h(x)\) into \(g(x)\). This results in \(g[h(x)] = g(2x + 5)\).

Step 3: Apply \(g(x)\) to \(h(x)\) result

Substitute \(x\) in \(g(x) = -x + 3\) with \(2x + 5\). This gives \(g(2x + 5) = -(2x + 5) + 3\). Simplify it to find the composite function: \(g(2x + 5) = -2x - 5 + 3 = -2x - 2\).

Step 4: Substitute for \(g(x)\) in \(h[g(x)]\)

Since \(g(x) = -x + 3\), substitute \(g(x)\) into \(h(x)\). This results in \(h[g(x)] = h(-x + 3)\).

Step 5: Apply \(h(x)\) to \(g(x)\) result

Substitute \(x\) in \(h(x) = 2x + 5\) with \(-x + 3\). This gives \(h(-x + 3) = 2(-x + 3) + 5\). Simplify it to find the composite function: \(h(-x + 3) = -2x + 6 + 5 = -2x + 11\).

Key Concepts

Function Notation

Function notation is a way to denote and evaluate functions in mathematics. Using function notation, equations and their operations become clearer and more structured. When we write a function in the form of \( f(x) \), it tells us that 'f' is a function of 'x'. This means that by substituting a value for 'x', the function produces a specific outcome, often represented by the variable 'y' or 'f(x).' Function notation allows us to easily perform operations involving one or more functions. For example:

• The function \( h(x) = 2x + 5 \) means that for any input 'x', the output is '2 times x plus 5'.
• The function \( g(x) = -x + 3 \) means for every input 'x', the result is 'negative x plus 3'.

Once you become familiar with function notation, it simplifies the process of working with complex expressions and makes it easier to follow operations like composition.

Function Composition

Function composition involves using one function as an input for another function. It is an operation that combines two functions into a single new function. This process is commonly denoted using notation such as \( g[h(x)] \) or \( h[g(x)] \), which means applying the function \( h \) to 'x' first, and then applying \( g \) to the result, and vice versa.
Understanding how to perform function composition is crucial for deepening your understanding of algebra and other advanced mathematics. Here's how it works:

• For \( g[h(x)] \): Calculate \( h(x) \) first (e.g., \( h(x) = 2x + 5 \)), then plug the result into \( g(x) \), which is \( g(2x + 5) = -2x - 5 + 3 = -2x - 2 \).
• For \( h[g(x)] \): Start with \( g(x) \) (e.g., \( -x + 3 \)), substitute into \( h(x) \), giving \( h(-x + 3) = 2(-x + 3) + 5 = -2x + 11 \).

By mastering these steps, you'll be able to solve problems involving composite functions with ease.

Algebraic Simplification

Algebraic simplification is the process of rewriting complex expressions into simpler or more manageable forms. It is a fundamental skill in algebra that helps in solving equations and significantly reduces errors. When simplifying, you should always aim for clarity and brevity, focusing on combining like terms and reducing expressions.
Here is a step-by-step guide to simplifying the expressions we have from our composite functions:

• When expanding \( -(2x + 5) + 3 \), apply distributive property: \( -1 \times (2x + 5) + 3 = -2x - 5 + 3 \). Then combine like terms to get \( -2x - 2 \).
• For \( 2(-x + 3) + 5 \), distribute \( 2 \) into the parenthesis: \( 2 \times (-x) + 2 \times 3 = -2x + 6 \), then add 5 to result in \( -2x + 11 \).

By consistently practicing algebraic simplification, you develop a strong foundation for tackling a wide variety of mathematical problems.