Problem 45 - College Algebra Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=\frac{8 x}{x-2}, g(x)=\frac{6}{x+3}$$

Step-by-Step Solution

Answer

\(f(x) + g(x)=\frac{8x^2+30x-12}{x^2+x-6}, f(x) - g(x)=\frac{8x^2+18x+12}{x^2+x-6}, f(x)g(x)= \frac{48x}{(x-2)(x+3)}, f(x)/g(x)= \frac{4x^2+12x}{3x-6}\). The domain for all these functions is \(x \nequal 2, -3\).

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

To perform the addition, sum the two given formulas together: \(f(x) + g(x)=\frac{8x}{x-2} + \frac{6}{x+3}\). To add these fractions, you need a common denominator, which is \((x-2)(x+3)\): \[f(x) + g(x)=\frac{8x(x+3) + 6(x-2)}{(x-2)(x+3)} = \frac{8x^2 + 24x + 6x -12}{x^2 + x -6} = \frac{8x^2+30x-12}{x^2+x-6}.\] The domain for this function is \(x \nequal 2, -3\).

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

To perform the subtraction, subtract \(g(x)\) from \(f(x)\): \(f(x) - g(x)=\frac{8x}{x-2} - \frac{6}{x+3}\). Similarly, to subtract these fractions, you need a common denominator, which is \((x-2)(x+3)\): \[f(x) - g(x)=\frac{8x(x+3) - 6(x-2)}{(x-2)(x+3)} = \frac{8x^2 + 24x - 6x +12}{x^2 + x -6} = \frac{8x^2+18x+12}{x^2+x-6}.\] The domain for this function is \(x \nequal 2, -3\).

Step 3: Multiplication \(f(x)g(x)\)

To perform the multiplication, multiply the two given formulas together: \(f(x)g(x)=\frac{8x}{x-2} \times \frac{6}{x+3} = \frac{48x}{(x-2)(x+3)}.\] The domain for this function is \(x \nequal 2, -3\).

Step 4: Division \(f(x)/g(x)\)

To perform the division, divide \(f(x)\) by \(g(x)\): \[f(x)/g(x)= \frac{8x/(x-2)}{6/(x+3)}\]. Which becomes, upon multiplying the numerator and denominator by \((x+3)\): \[f(x)/g(x)= \frac{8x(x+3)}{6(x-2)} = \frac{8x^2 + 24x}{6x - 12} = \frac{4x^2+12x}{3x-6}.\] The domain for this function is \(x \nequal 2, -3\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, arranged in terms of the powers of the variables. Each term is a product of a coefficient and a power of a variable. For example, in the expression \(8x^2 + 30x - 12\), the numbers 8, 30, and -12 are coefficients, and \(x^2\) and \(x\) are the variable terms.

• A polynomial can have one or more terms, and it is defined for all real numbers.
• The degree of a polynomial is determined by the highest power of the variable present in the expression.
• In the example of \(8x^2 + 30x - 12\), the degree is 2, since the highest power is \(x^2\).
Polynomials can be added, subtracted, multiplied and divided, though polynomial division is a bit more complex and sometimes does not result in another polynomial. In operations involving polynomials, it is important to combine like terms. Addition, subtraction, and multiplication of polynomials are relatively straightforward, as they involve distributing and combining terms according to their exponents.

Rational Functions

Rational functions are expressions made up of the ratio of two polynomials. For example, the functions \(f(x) = \frac{8x}{x-2}\) and \(g(x) = \frac{6}{x+3}\) are rational functions because they are fractions with polynomials in both the numerator and denominator.

• The expression in the numerator and the denominator independently acts like a polynomial.
• Rational functions are more complex because they can have asymptotes, which are lines that the graph of the function approaches but never touches.
• Finding common denominators is essential when adding or subtracting rational functions.

Performing operations on rational functions involves careful manipulation to maintain the integrity of the expression. For example, when adding \(f(x)\) and \(g(x)\), we need a common denominator. This involves developing an understanding of how the polynomial expressions in the numerator and denominator interact.

Function Domains

The domain of a function informs us about which values of \(x\) are allowed in an expression. For rational functions, this means identifying values that make the denominator zero, because division by zero is undefined.

• In general, a polynomial function has the domain of all real numbers.
• However, for rational functions, any value that makes the denominator zero must be excluded from the domain.
• For instance, for \(f(x) = \frac{8x}{x-2}\), the domain excludes \(x = 2\) because it makes the denominator zero.
• Similarly, \(g(x) = \frac{6}{x+3}\) excludes \(x = -3\).

When determining the domain of combined functions such as \(f+g\) or \(f/g\), it is important to consider the domain restrictions of each function involved. Always check the final expression, because the operations might introduce new restrictions not present in the original functions.