Problem 4 - Algebra for College Students

Question

\(f(x)=2 x^{2}-3 x+5, \quad g(x)=x^{2}-4\)

Step-by-Step Solution

Answer

\(f(x) + g(x) = 3x^2 - 3x + 1\), \(f(x) - g(x) = x^2 - 3x + 9\).

Step 1: Understand the Functions

We are given two functions: 1. \( f(x) = 2x^2 - 3x + 5 \) 2. \( g(x) = x^2 - 4 \) These functions can be used for various operations such as addition, subtraction, multiplication, or composition.

Step 2: Calculate \(f(x) + g(x)\)

To find \(f(x) + g(x)\), add the terms of both functions: \[ f(x) + g(x) = (2x^2 - 3x + 5) + (x^2 - 4) \] Combine like terms: \[ f(x) + g(x) = 2x^2 + x^2 - 3x + 5 - 4 = 3x^2 - 3x + 1 \] So, \( f(x) + g(x) = 3x^2 - 3x + 1 \).

Step 3: Calculate \(f(x) - g(x)\)

To find \(f(x) - g(x)\), subtract the terms of \(g(x)\) from \(f(x)\): \[ f(x) - g(x) = (2x^2 - 3x + 5) - (x^2 - 4) \] Simplify by distributing and combining like terms: \[ f(x) - g(x) = 2x^2 - x^2 - 3x + 5 + 4 = x^2 - 3x + 9 \] So, \( f(x) - g(x) = x^2 - 3x + 9 \).

Key Concepts

Function Addition

When working with polynomial functions, one common operation is function addition. This involves adding two functions together. To illustrate, consider two functions: \( f(x) = 2x^2 - 3x + 5 \) and \( g(x) = x^2 - 4 \). The aim is to find a new function that represents the sum of these two functions.

To perform function addition, each term from the first function \( f(x) \) is added to its corresponding term in the second function \( g(x) \). So, \( f(x) + g(x) \) becomes:

• Add the terms with \( x^2 \): \((2x^2) + (x^2) = 3x^2\)
• Add the terms with \( x \): \(-3x\), since there is no \( x \) term in \( g(x) \), it remains \(-3x\)
• Add the constant terms: \( 5 + (-4) = 1 \)

Combining these results gives us the new function \( f(x) + g(x) = 3x^2 - 3x + 1 \). This operation is useful in various algebraic applications where combining expressions is essential.

Function Subtraction

Function subtraction is another important operation in dealing with polynomial functions. It involves finding the difference between two functions. Let's continue with our same example functions: \( f(x) = 2x^2 - 3x + 5 \) and \( g(x) = x^2 - 4 \).

To subtract one function from another, we subtract each corresponding term from \( g(x) \) out of \( f(x) \). To do this, \( f(x) - g(x) \) will be:

• Subtract the \( x^2 \) terms: \((2x^2) - (x^2) = x^2\)
• Subtract the \( x \) terms: Again, there is no matching \( x \) term in \( g(x) \), leaving \(-3x\)
• Subtract the constant terms: \(5 - (-4) = 9\)

Putting it all together, the resulting function is \( f(x) - g(x) = x^2 - 3x + 9 \). Subtracting functions is beneficial when you need to find how two functions differ or oppose one another in particular contexts.

Combining Like Terms

Combining like terms is a crucial aspect of simplifying polynomial expressions. It involves grouping and adding or subtracting terms that have the same variable raised to the same power. This process is seen in both function addition and subtraction.

When combining like terms, pay attention to the following steps:

• Identify all like terms in the expression. For example, terms such as \(2x^2\) and \(x^2\) can be combined, as can constant numbers like \(5\) and \(-4\).
• Sum or subtract these like terms by combining their coefficients. For example, \(2x^2 + x^2 = 3x^2\).
• Keep terms that have no like partners intact, like the \(-3x\) term in our examples.

Combining like terms simplifies expressions and makes them easier to work with. It ensures that results are in their simplest form, aiding in interpretations and further mathematical operations.