Problem 1 - Contemporary Precalculus

Question

Find \((f+g)(x),(f-g)(x),\) and \((g-f)(x)\) $$f(x)=-3 x+2, \quad g(x)=x^{3}$$

Step-by-Step Solution

Answer

The resulting functions are: 1. $(f+g)(x) = x^3 - 3x + 2$ 2. $(f-g)(x) = -x^3 - 3x + 2$ 3. $(g-f)(x) = x^3 + 3x - 2$

Step 1: Find \((f+g)(x)\)

We need to add the functions \(f(x)\) and \(g(x)\) to find \((f+g)(x)\). Using the given functions, \(f(x) = -3x + 2\) and \(g(x) = x^3\), we get the following expression: $$(f+g)(x) = (-3x + 2) + (x^3)$$ Now, we simplify the expression: $$(f+g)(x) = x^3 - 3x + 2$$ So, \((f+g)(x)=x^3 - 3x + 2\).

Step 2: Find \((f-g)(x)\)

Next, we need to subtract the function \(g(x)\) from \(f(x)\) to find \((f-g)(x)\). Using the given functions, we get the following expression: $$(f-g)(x) = (-3x + 2) - (x^3)$$ Now, we simplify the expression: $$(f-g)(x) = -x^3 - 3x + 2$$ So, \((f-g)(x)=-x^3 - 3x + 2\).

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

Finally, we need to subtract the function \(f(x)\) from \(g(x)\) to find \((g-f)(x)\). Using the given functions, we get the following expression: $$(g-f)(x) = (x^3) - (-3x + 2)$$ Now, we simplify the expression: $$(g-f)(x) = x^3 - (-3x) - 2$$ $$(g-f)(x) = x^3 + 3x - 2$$ So, \((g-f)(x) = x^3 + 3x - 2\). In conclusion, we found that \((f+g)(x)=x^3 - 3x + 2\), \((f-g)(x)=-x^3 - 3x + 2\), and \((g-f)(x) = x^3 + 3x - 2\).

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, constructed using only operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions are found in various mathematical contexts and can take on different forms, depending on the degree of the polynomial.
A polynomial's degree is the highest power of the variable within the polynomial. For example, the function \(g(x) = x^3\) is a polynomial of degree 3 because the variable \(x\) is raised to the power of 3. Another example is \(f(x) = -3x + 2\), a polynomial of degree 1, known as a linear polynomial.
Polynomials can be simple, like \(x^2\), or more complex, involving several terms, such as \(x^3 - 3x + 2\). The structure of polynomial functions allows them to be combined or manipulated through addition and subtraction, among other operations.

Function Addition

Function addition involves creating a new function by adding together two or more functions. When you add functions, you combine like terms and simplify. It works similarly to algebra, where you align terms based on their degrees and then perform simple addition.
Let's consider the functions \(f(x) = -3x + 2\) and \(g(x) = x^3\). To find \((f+g)(x)\), you add the functions:

• Align terms based on power: the term \(x^3\) doesn't have a counterpart in \(f(x)\), so it remains the same.
• The linear terms from both functions are \(-3x\) and +0x (because there is no linear term in \(g(x)\)). Combine these terms to get \(-3x\).
• The constant terms are +2 from \(f(x)\) and 0 from \(g(x)\), resulting in +2.

Thus, the function \((f+g)(x)\) simplifies to \(x^3 - 3x + 2\).
Function addition is a powerful tool for combining polynomial functions, enabling you to analyze the combined effects of different algebraic expressions.

Function Subtraction

Function subtraction is similar to addition, but instead, you subtract one function from another. This operation can be used to find the difference between the outcomes of two functions. The method involves subtracting each corresponding term of the functions.
The exercise showed a few examples, such as \((f-g)(x)\) and \((g-f)(x)\). To calculate \((f-g)(x)\), you handle it this way:

• Subtract the entire function \(g(x)\) from \(f(x)\), starting with the highest degree terms.
• The high degree term result: \(-x^3\) from subtracting \(x^3\) (from \(g(x)\)) from \(0x^3\) (since \(f(x)\) lacks this term).
• The linear terms yield \(-3x\), and the constant results in +2.

Hence, \((f-g)(x)\) gives you \(-x^3 - 3x + 2\).
Similarly, reversing the subtraction as in \((g-f)(x)\) yields \(x^3 + 3x - 2\).
Function subtraction helps in understanding the relationship between different polynomials and is a fundamental operation in algebraic manipulation.