Problem 3 - Beginning and Intermediate Algebra

Question

For each pair of functions, find a) \((f+g)(x)\) b) \((f+g)(5), \text { c) }(f-g)(x), \text { and } \mathrm{d})(f-g)(2)\) $$f(x)=4 x^{2}-7 x-1, g(x)=x^{2}+3 x-6$$

Step-by-Step Solution

Answer

a) \((f+g)(x) = 5x^2 - 4x - 7\) b) \((f+g)(5) = 98\) c) \((f-g)(x) = 3x^2 - 10x + 5\) d) \((f-g)(2) = -3\)

Step 1: Finding (f+g)(x)

To find the expression (f+g)(x), add f(x) and g(x) for all x. \[ (f+g)(x) = f(x) + g(x)\] \[ =(4x^2 - 7x - 1) + (x^2 + 3x - 6)\]

Step 2: Simplify (f+g)(x)

Combine each of the like terms in Step 1. \[ (f+g)(x) = (4x^2 + x^2) + (-7x + 3x) + (-1 - 6)\] \[ (f+g)(x) = 5x^2 - 4x - 7\]

Step 3: Finding (f+g)(5)

Plug in 5 for x in the simplified (f+g)(x) expression from Step 2. \[ (f+g)(5) = 5(5)^2 - 4(5) - 7\]

Step 4: Calculate (f+g)(5)

Evaluate the expression found in Step 3. \[ (f+g)(5) = 5(25) - 20 - 7\] \[ (f+g)(5) = 125 - 20 - 7\] \[ (f+g)(5) = 98\]

Step 5: Finding (f-g)(x)

To find the expression (f-g)(x), subtract g(x) from f(x) for all x. \[ (f - g)(x) = f(x) - g(x)\] \[ =(4x^2 - 7x - 1) - (x^2 + 3x - 6)\]

Step 6: Simplify (f-g)(x)

Distribute the negative sign and combine like terms in Step 5. \[ (f - g)(x) = (4x^2 - x^2) + (-7x - 3x) + (-1 + 6)\] \[ (f - g)(x) = 3x^2 - 10x + 5\]

Step 7: Finding (f-g)(2)

Plug in 2 for x in the simplified (f-g)(x) expression from Step 6. \[ (f-g)(2) = 3(2)^2 - 10(2) + 5\]

Step 8: Calculate (f-g)(2)

Evaluate the expression found in Step 7. \[ (f-g)(2) = 3(4) - 20 + 5\] \[ (f-g)(2) = 12 - 20 + 5\] \[ (f-g)(2) = -3\] So, the expressions are: a) \((f+g)(x) = 5x^2 - 4x - 7\) b) \((f+g)(5) = 98\) c) \((f-g)(x) = 3x^2 - 10x + 5\) d) \((f-g)(2) = -3\)

Key Concepts

Function Operations

Understanding the basics of function operations is crucial when working with polynomial functions. Functions such as \(f(x)\) and \(g(x)\) can be added, subtracted, multiplied, or divided. In this exercise, we focus on addition and subtraction.

• Addition of Functions: When you add functions, you essentially add each corresponding term. For two functions \(f(x)\) and \(g(x)\), the sum is \((f+g)(x) = f(x) + g(x)\).
• Subtraction of Functions: Similarly, subtracting functions involves subtracting each term of the second function from the corresponding term of the first. This is expressed as \((f-g)(x) = f(x) - g(x)\).

Performing these operations requires careful attention to the signs of each term, especially when subtracting, to ensure accurate simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In this exercise, we are given two specific polynomials:

• \(f(x) = 4x^{2}-7x-1\)
• \(g(x) = x^{2}+3x-6\)

These are polynomials, which are a type of algebraic expression. They consist of terms, raised to non-negative integer powers, that are summed together. The highest degree term (term with the largest exponent, in this case, \(x^2\) for \(g(x)\) and \(4x^2\) for \(f(x)\)) dictates the degree of the polynomial, which reflects its growth rate and general shape.

Simplifying Expressions

Simplifying expressions is a key algebraic skill. It involves combining like terms to create a more compact, easy-to-understand expression.

• Combining Like Terms: In the subtraction \((f - g)(x)\) and addition \((f+g)(x)\), terms with the same powers of \(x\) are combined. For instance, \(4x^2 + x^2 = 5x^2\), and \(-7x + 3x = -4x\).
• Importance of Sign Operations: Especially in subtraction, distributing the negative sign correctly (as in \( (f-g)(x) = (4x^2 - x^2) + (-7x - 3x) + (-1 + 6)\)) is crucial for correct results.
• Simplification Process: Ultimately, simplifying helps in easily evaluating the expression for specific values or solving equations. After simplification, evaluating at specific points (like finding \((f+g)(5)\) or \((f-g)(2)\)) becomes straightforward.

Taking each step methodically and rechecking signs and operations ensures accuracy and enhances your understanding of polynomial behavior.