Problem 77 - Algebra 2

Question

Find \(g[h(x)]\) and \(h[g(x)]\) $$ \begin{array}{l}{h(x)=2 x-1} \\ {g(x)=x-5}\end{array} $$

Step-by-Step Solution

Answer

\(g[h(x)] = 2x - 6\) and \(h[g(x)] = 2x - 11\).

Step 1: Substitute h(x) into g(x)

To find \(g[h(x)]\), replace \(x\) in \(g(x) = x - 5\) with \(h(x) = 2x - 1\). You will have: \[ g[h(x)] = (2x - 1) - 5. \]

Step 2: Simplify g[h(x)]

Simplify the expression \((2x - 1) - 5\). Calculate as follows: \[ 2x - 1 - 5 = 2x - 6. \] This gives us \(g[h(x)] = 2x - 6\).

Step 3: Substitute g(x) into h(x)

To find \(h[g(x)]\), replace \(x\) in \(h(x) = 2x - 1\) with \(g(x) = x - 5\). You will have: \[ h[g(x)] = 2(x - 5) - 1. \]

Step 4: Simplify h[g(x)]

Simplify the expression \(2(x - 5) - 1\). Distribute and combine like terms: \[ 2(x - 5) = 2x - 10 \] and then \[ 2x - 10 - 1 = 2x - 11. \] This gives us \(h[g(x)] = 2x - 11\).

Key Concepts

Algebraic Functions

Understanding algebraic functions is crucial for addressing problems related to function composition. Simply put, an algebraic function is an expression that uses algebraic operations such as addition, subtraction, multiplication, division, and raising to a power, involving one or more variables.
In the exercise given, we deal with two basic algebraic functions:

• Function h (x), which represents the equation: \( h(x) = 2x - 1 \)
• Function g (x), representing the equation: \( g(x) = x - 5 \)

These functions can be visualized as operations that take an input x and produce an output by transforming it following the rules given in each function. Recognizing how these transformations behave through algebraic functions makes tackling problems like function composition a breeze.

Substitution Method

The substitution method is a powerful technique in algebra used to solve equations and perform operations such as function composition. In this context, it involves replacing a variable with another expression. This process allows you to transfer outputs between functions seamlessly.
To understand it better, let's look at how we composed two functions from the exercise:

• **Substitution for \( g[h(x)] \)**: Here, we replace every x in the function \( g(x) = x - 5 \) with the whole expression for \( h(x) = 2x - 1 \). This leads to a new expression \( g[2x - 1] \) which simplifies to \( 2x - 6 \).
• **Substitution for \( h[g(x)] \)**: Similarly, this involves replacing x in \( h(x) = 2x - 1 \) with the expression for \( g(x) = x - 5 \), to get \( h[x - 5] \) and simplify it to \( 2x - 11 \).

Through substitution, you essentially operate one function within another, resulting in an entirely new function that reflects the properties of both original functions.

Simplification Process

The simplification process is essential in making complex expressions more manageable and understandable. It involves eliminating unnecessary terms and combining like terms to reduce the expression to its simplest form.
In our exercise, the simplification process was applied after each substitution step:

• For \( g[h(x)] = (2x - 1) - 5 \), simplification transformed it into a neat expression, \( 2x - 6 \), by combining the constants \(-1\) and \(-5\).
• Similarly, in \( h[g(x)] = 2(x - 5) - 1 \), the distribution of the 2 over the parenthesis resulted in \( 2x - 10 \), and further combining that with \(-1\) produced the simple expression \( 2x - 11 \).

A careful simplification ensures that the final expression is free from redundancies, making it simpler for further mathematical operations or interpretations.