Problem 6 - College Algebra

Question

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=-3 x^{2}+x\) and \(g(x)=5,\) find \(f+g\) \(f-g, f g\) , and \(\frac{f}{g}\)

Step-by-Step Solution

Answer

The domain for all resulting functions is \((-\infty, \infty)\).

Step 1: Define the Functions

First, we define the given functions: \(f(x) = -3x^2 + x\) and \(g(x) = 5\). These definitions will help us in performing the operations.

Step 2: Calculate \(f+g\)

To find \(f+g\), we simply add the two functions together: \((f+g)(x) = (-3x^2 + x) + 5 = -3x^2 + x + 5\).

Step 3: Determine the Domain of \(f+g\)

The domain of \(f(x)\) and \(g(x)\) are both all real numbers. Since no division or root expressions are involved, the domain of \(f+g\) is all real numbers, which is \((-\infty, \infty)\) in interval notation.

Step 4: Calculate \(f-g\)

To find \(f-g\), subtract the second function from the first: \((f-g)(x) = (-3x^2 + x) - 5 = -3x^2 + x - 5\).

Step 5: Determine the Domain of \(f-g\)

Similarly to \(f+g\), since \(f\) and \(g\) have domains of all real numbers without restrictions, \(f-g\) also has a domain of \((-\infty, \infty)\).

Step 6: Calculate \(f\cdot g\)

Multiply the functions: \((f \cdot g)(x) = (-3x^2 + x) \cdot 5 = -15x^2 + 5x\).

Step 7: Determine the Domain of \(f\cdot g\)

Products of polynomials do not introduce any domain restrictions beyond those already present, so the domain of \(f \cdot g\) is \((-\infty, \infty)\).

Step 8: Calculate \(\frac{f}{g}\)

Divide the first function by the second: \(\left(\frac{f}{g}\right)(x) = \frac{-3x^2 + x}{5} = -\frac{3}{5}x^2 + \frac{1}{5}x\).

Step 9: Determine the Domain of \(\frac{f}{g}\)

Since \(g(x) = 5\) is non-zero for all real numbers, the domain of \(\frac{f}{g}\) is also unrestricted, \((-\infty, \infty)\).

Key Concepts

Interval Notation

When talking about the domain of functions, interval notation is a handy way to express the set of possible input values, or "x-values," for which a function is defined.
Interval notation uses brackets and parentheses to denote intervals on the real number line:

• "(" or ")" represents that an endpoint is not included in the interval.
• "[" or "]" indicates inclusion of the endpoint.

For example, the interval \((-\infty, \infty)\), which we use in our solution, shows that a function's domain includes all real numbers.
In contrast, an interval like \([0, 5]\) would describe a domain that only includes numbers from 0 to 5, inclusive of both endpoints.

Polynomial Functions

Polynomial functions are made from terms that are non-negative integer powers of one or more variables. A polynomial function might look like this: \(f(x) = -3x^2 + x\).
Characteristics of polynomial functions include:

• They are smooth and continuous, meaning there are no breaks, gaps, or holes in their graphs.
• The domain of any polynomial function is typically all real numbers, \((-\infty, \infty)\). This is because polynomials are defined everywhere unless specified otherwise.
• They can be easily evaluated for any input value.

Thus, in our exercise, both \(f(x)\) and \(g(x)\), although \(g(x)\) is constant, share this common domain.

Function Operations

Function operations allow us to combine functions in various ways, such as addition, subtraction, multiplication, and division. Each of these operations can impact the domain of the resulting function.
In our exercise:

• For \(f+g\) and \(f-g\), we add or subtract corresponding function values, resulting in another function whose domain includes all inputs from the contributing functions.
• When multiplying functions like \(f \/\cdot\/ g\), the domain is still all real numbers, similar to the contributing polynomial functions.
• For dividing \(f/g\), we must ensure the denominator is not zero across the domain, but since \(g(x) = 5\) never hits zero, the domain remains \((-\infty, \infty)\).

Remember, these operations allow for versatile and creative function manipulation as long as domain considerations are respected.

Rational Functions

Rational functions are expressed as the quotient of two polynomials, like \(\frac{f(x)}{g(x)}\). These can introduce domain restrictions because the denominator cannot be zero.
For example, in our solved problem, we work with \(\frac{f}{g}\). Since \(g(x) = 5\) is never zero, there is no additional domain restriction beyond all real numbers.
Some key points to remember about rational functions include:

• Always check the denominator to ensure it does not equal zero for any x-value.
• If so, exclude those x-values from the domain, adjusting the interval notation accordingly.
• Evaluate potential domain restrictions when dealing with more complex rational functions.

In simpler terms, rational functions require careful consideration of their defined intervals to avoid undefined expressions.