Problem 69 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=2 x-1\) and \(g(x)=x^{2}+x-2,\) find \((g \circ f)(x)\)

Step-by-Step Solution

Answer

(g \circ f)(x) = 4x^2 - 2x - 2.

Step 1: Understand the Problem

The problem requires finding the composition of two functions, which means you need to find \(g(f(x))\).

Step 2: Identify Inner Function

Identify the inner function which is \(f(x) = 2x - 1\).

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

Substitute \(f(x)\) into \(g(x)\). The expression for \(g(x)\) is \(g(x) = x^2 + x - 2\), so you will replace every \(x\) in \(g(x)\) with \(2x - 1\).

Step 4: Simplify the Expression

Simplify \[g(f(x)) = (2x - 1)^2 + (2x - 1) - 2\]. Expand and combine like terms:

Step 5: Expand \( (2x - 1)^2 \)

Start by expanding \( (2x - 1)^2 \) as follows: \[ (2x - 1)^2 = 4x^2 - 4x + 1\]

Step 6: Combine All Terms

Combine all terms: \[g(f(x)) = 4x^2 - 4x + 1 + 2x - 1 - 2 = 4x^2 - 2x - 2\]

Key Concepts

Understanding Function Composition

When working with two functions, such as \(f(x)\) and \(g(x)\), function composition involves creating a new function by substituting one function into the other. This is written as \(g(f(x))\). It helps to think of this as plugging the output of \(f(x)\) into \(g(x)\). For example, if you have \(f(x) = 2x - 1\) and \(g(x) = x^2 + x - 2\), you need to find what \(g(f(x))\) is.

Substitution in Function Composition

Substitution is a key step in function composition. It involves replacing the variable in one function with an expression from another function. For instance, if you have \(f(x) = 2x - 1\), you will take this entire expression and substitute it into every \(x\) in \(g(x)\). When substituting \(f(x)\) into \(g(x)\), we get:

• Original \(g(x)\): \(x^2 + x - 2\)
• After substitution: \(g(f(x)) = (2x - 1)^2 + (2x - 1) - 2\)

Simplifying Expressions after Substitution

After substitution, it's crucial to simplify the resulting expression. Simplifying here means performing algebraic operations to combine like terms and make the expression more manageable. From the previous step, we have:

• Initial expression: \((2x - 1)^2 + (2x - 1) - 2\)
• Expanded form: \((2x - 1)^2 = 4x^2 - 4x + 1\)

Now combine all terms to simplify the expression:

• \(g(f(x)) = 4x^2 - 4x + 1 + 2x - 1 - 2\)
• Final simplified form: \(4x^2 - 2x - 2\)

Identifying the Inner Function

The inner function is the function that you substitute into the other. In \(g(f(x))\), \(f(x)\) is the inner function. This means whenever you see \(x\) in \(g(x)\), you will replace it with the entire expression of \(f(x)\). For our exercise with \(f(x) = 2x - 1\) and \(g(x) = x^2 + x - 2\), we set:

• Inner function (\(f(x)\)): \(2x - 1\)
• Outer function (\(g(x)\)): \(x^2 + x - 2\)

Expanding Binomials

Expanding binomials is a necessary skill when simplifying the expression after substitution. When you have a binomial squared, like \((2x - 1)^2\), you expand it using the formula \((a - b)^2 = a^2 - 2ab + b^2\). In this example:

• a = 2x, b = 1
• Expand using the formula: \((2x - 1)^2 = 4x^2 - 4x + 1\)

Expanding correctly will help in combining like terms accurately in the final step.