Problem 46 - College Algebra Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=\frac{9 x}{x-4}, g(x)=\frac{7}{x+8}$$

Step-by-Step Solution

Answer

The functions are as follows: \(f + g =\frac{9x^2+19x+56}{(x-4)(x+8)}\), \(f - g=\frac{9x^2-105x+28}{(x-4)(x+8)}\), \(fg = \frac{63x}{(x-4)(x+8)}\) and \(f/g = \frac{9x^2}{7(x-4)}\). The domain excludes the values \(x = 4\) and \(x = -8\).

Step 1: Find the sum, f + g

To sum the two functions, simply add the functions together. Thus, \(f+g= \frac{9x}{x-4}+\frac{7}{x+8}\). To simplify, find a common denominator and simplify to yield the final result.

Step 2: Find the difference, f - g

Subtract the second function from the first function to get the difference, \(f - g = \frac{9x}{x-4} - \frac{7}{x+8}\). Like in step one, find a common denominator and simplify to yield the result.

Step 3: Find the product, fg

Multiply the functions together to obtain the product function, \(fg = \left(\frac{9x}{x-4}\right)\left(\frac{7}{x+8}\right)\). Simply multiply the numerators together to get a new numerator and the denominators together to form a new denominator to obtain the final result.

Step 4: Find the quotient, f/g

Divide the first function by the second to obtain the quotient, \(f/g = \frac{\frac{9x}{x-4}}{\frac{7}{x+8}}\). To simplify this, multiply the first function by the reciprocal of the second function.

Step 5: Determine the domain

The domain of these functions will include all artfph numbers, except for the values that make the denominator equal to zero. Set the denominator of each function to zero. Solve for x to find out which values must be excluded from the domain.

Key Concepts

Operations on Functions

Understanding the operations on functions is a fundamental aspect in college algebra, particularly when dealing with complex function expressions.

When adding \texttt{f+g}\texttt{\(f+g\)} or subtracting \texttt{f-g}\texttt{\(f-g\)} functions, you essentially combine the values produced by each function. Ensure you find a common denominator if the functions are rational, as it allows you to combine the fractions into a single expression. For \texttt{f+g}\texttt{\(f+g\)}, you add the numerators after this alignment, and for \texttt{f-g}\texttt{\(f-g\)}, you subtract them.

Multiplying functions, indicated as \texttt{fg}\texttt{\(fg\)}, entails multiplying the numerators together to find a new numerator, and doing the same with the denominators for a new denominator. The operation reflects the scaling or stretching effect one function has on the other.

Dividing functions, shown by \texttt{f/g}\texttt{\(f/g\)}, is akin to multiplying by the reciprocal. Essentially, you flip the second function, creating a reciprocal, and then proceed as you would in multiplication. This operation can reveal how one function inversely affects the other.

• For addition and subtraction, find a common denominator to combine functions.
• For multiplication and division, multiply or divide, respectively, the numerators and denominators.

Function Domain

The domain of a function defines the set of all possible input values, which the function can accept without causing any mathematical errors. For example, with rational functions, the domain is all real numbers except those that make the denominator equal to zero, since division by zero is undefined.

To identify the domain, you would set the denominator equal to zero and solve for \(x\). In the given exercise, the functions \(f(x)=\frac{9x}{x-4}\) and \(g(x)=\frac{7}{x+8}\) both contain denominators that when set to zero, give you the values to exclude from the domain: \(x-4=0\) yields \(x=4\) and \(x+8=0\) yields \(x=-8\). Thus, the domain of f includes all real numbers except 4, and the domain of g excludes -8.

• Identify non-permissible values that make denominators zero to determine exclusions from the domain.
• Express the domain typically in interval notation, real numbers except restricted values.

Remember, any values that cause the denominator to become zero are not a part of the function's domain.

Simplifying Rational Expressions

Simplifying rational expressions is a key skill in algebra that involves rewriting expressions into their simplest form. This often requires factoring both the numerator and denominator and canceling out common factors. However, when there are no common factors, simplifying can involve finding a common denominator if adding or subtracting rational expressions is in order.

In the provided exercise, when summing or subtracting the functions, since the denominators are different, you will first find a common denominator to effectively combine the terms. Simplification may then include reducing the resulting fraction by canceling any common factors between the numerator and denominator.

• Factor both the numerator and denominator if possible.
• When combining rational expressions, first find a common denominator then simplify.
• Always check for and simplify common factors after operations.

Taking careful steps to simplify rational expressions ensures a clear and reduced form, making subsequent mathematical operations simpler and error-free.