Problem 9 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-16,\) for the given functions \(f\) and \(g,\) find each composite function and identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \(\left(\frac{f}{g}\right)(x)\) $$f(x)=x-3 ; g(x)=x^{2}+1$$

Step-by-Step Solution

Answer

(f+g)(x) = \(x²+x-2\) with domain \(R\), (f-g)(x) = \(1-x-x²\) with domain \(R\), (fg)(x) = \(x³-x²-2x+3\) with domain \(R\), \(\left(\frac{f}{g}\right)(x) = \frac{x-3}{x²+1}\) with domain \(R\).

Step 1: Calculation of (f+g)(x)

First, perform the addition operation of the two functions f(x)=x-3 and g(x)=x²+1: \((f+g)(x)=f(x)+g(x)=(x-3)+(x²+1)=x²+x-2.

Step 2: Domain of (f+g)(x)

The result is a polynomial, there are no restrictions on x and thus the domain would be \(R\), the set of all real numbers.

Step 3: Calculation of (f-g)(x)

Next, perform the subtraction operation - subtract g(x) from f(x): \((f-g)(x)=f(x)-g(x)=(x-3)-(x²+1)=1-x-x².

Step 4: Domain of (f-g)(x)

Similar to the addition, the result is a polynomial, there are no restrictions on x and thus the domain would be \(R\), the set of all real numbers.

Step 5: Calculation of (fg)(x)

Now, proceed to the multiplication operation of the two functions f(x)=x-3 and g(x)=x²+1: \((fg)(x)=f(x)g(x)=(x-3)(x²+1)=x³-x²-2x+3.

Step 6: Domain of (fg)(x)

As with the previous operations, the result is also a polynomial with no restrictions on x, hence the domain would be \(R\), the set of all real numbers.

Step 7: Calculation of \(\left(\frac{f}{g}\right)(x)\)

Finally, perform the division operation - divide f(x) by g(x): \(\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{x-3}{x²+1}.

Step 8: Domain of \(\left(\frac{f}{g}\right)(x)\)

In case of a division of functions, we have to ensure that the denominator is not equal to zero. Now, g(x)=x²+1 will never be zero for any x in \(R\), so the domain for this operation is also \(R\), the set of all real numbers.

Key Concepts

Function Operations

Function operations involve combining functions using basic mathematical operations like addition, subtraction, multiplication, and division. Each of these operations has specific procedures that can affect the resulting function and its domain. For example, when adding or subtracting two functions such as \(f(x) = x - 3\) and \(g(x) = x^2 + 1\), you simply add or subtract their expressions. In this case,

• \((f+g)(x) = (x - 3) + (x^2 + 1) = x^2 + x - 2\)
• \((f-g)(x) = (x - 3) - (x^2 + 1) = -x^2 + x - 4\)

Similarly, for multiplication, you multiply the expressions:

• \((fg)(x) = (x - 3)(x^2 + 1) = x^3 - x^2 - 2x + 3\)

Finally, for division, ensure you write the functions as a fraction, taking care that the denominator does not become zero:

• \(\left(\frac{f}{g}\right)(x)=\frac{x-3}{x^2+1}\)

Mastering these operations allows you to manipulate and analyze functions efficiently.

Domain of Functions

The domain of a function is the set of all possible input values (x-values) that the function can accept without causing undefined operations, like division by zero. When performing function operations, it's crucial to determine the domain of the result.
If the resulting expression is a polynomial, like

• \(x^2 + x - 2\)
• \(-x^2 + x - 4\)
• \(x^3 - x^2 - 2x + 3\)

the domain is \(\mathbb{R}\) because polynomials are defined for all real numbers and do not have restrictions.
However, for division problems, such as \(\left(\frac{f}{g}\right)(x)=\frac{x-3}{x^2+1}\), the primary concern is the denominator. Since \(x^2 + 1\) is always positive (and never zero) for any real number x, the domain remains \(\mathbb{R}\).
Understanding the domain helps prevent errors and contradictions in your mathematical work.

Polynomials

Polynomials are algebraic expressions that involve sums and products of variables and coefficients. They are one of the most common forms of functions in mathematics. The degree of a polynomial indicates the highest power of the variable in the expression.
For example:

• In \(x^2 + x - 2\), the degree is 2.
• In \(-x^2 + x - 4\), the degree is 2.
• In \(x^3 - x^2 - 2x + 3\), the degree is 3.

Polynomials can be combined through operations like addition, subtraction, and multiplication without affecting their continuity or domain.
When performing these operations, the result is also a polynomial.
Polynomials are smooth and continuous functions, defined for all real numbers, which makes understanding them crucially important for working with different types of mathematical problems.