Problem 73 - Intermediate Algebra

Question

Let \(f(x)=x^{2}-1\) and \(g(x)=x^{2}-4 .\) Find each function and give its domain. $$ f-g $$

Step-by-Step Solution

Answer

The function \( f-g \) is constant \( 3 \) with domain \( \mathbb{R} \).

Step 1: Identify the Functions

We have two functions given: \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \). These are both quadratic functions.

Step 2: Subtract the Functions

The task requires us to find \( f-g \). To do this, we subtract \( g(x) \) from \( f(x) \): \[(f-g)(x) = f(x) - g(x) = (x^2 - 1) - (x^2 - 4)\] Simplifying, we get:\[(f-g)(x) = x^2 - 1 - x^2 + 4 = 3\] Thus, \( (f-g)(x) = 3 \).

Step 3: Find the Domain

Both \( f(x) \) and \( g(x) \) are quadratic functions, which are defined for all real numbers because any real number can be squared and subtracted by a constant. Since there are no restrictions like division by zero or square roots of negative numbers, the domain for \( f(x) \), \( g(x) \), and thus \( f-g \) is all real numbers, \( \mathbb{R} \).

Key Concepts

Function Subtraction

Function subtraction is a process where you take one function and subtract another from it. For instance, if you have two functions, such as \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \), you can create a new function by subtracting \( g(x) \) from \( f(x) \). This is denoted by \((f-g)(x)\). Here’s the step-by-step:

• First, write the subtraction as \( (f-g)(x) = f(x) - g(x) \).
• Substitute the expressions for \( f(x) \) and \( g(x) \): \( (f-g)(x) = (x^2 - 1) - (x^2 - 4) \).
• Simplify the expression by combining like terms: \( (f-g)(x) = x^2 - 1 - x^2 + 4 \).
• The \( x^2 \) terms cancel out, leaving you with \( (f-g)(x) = 3 \).

In this example, the result of subtracting \( g(x) \) from \( f(x) \) simplifies to a constant value, 3. This means that for any input \( x \), the result of \( (f-g)(x) \) will always be 3.

Domain of a Function

The domain of a function is the set of all possible input values (often \( x \)) that the function can accept without causing any mathematical errors. For many basic functions, the domain is all real numbers. However, there are cases where restrictions apply, such as:

• Division by zero, which is undefined.
• Square roots of negative numbers, which aren't real-valued.

For our functions \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \), both are quadratic functions. Quadratic functions do not have any fractions, roots, or logarithms that might restrict the domain. Therefore, the domain for both functions is all real numbers. Since we are performing subtraction with these two quadratic functions in \( f-g \), the domain remains all real numbers, \( \mathbb{R} \). This means that you can substitute any real number for \( x \) in these operations without encountering any issues.

Real Numbers

Real numbers include all the numbers on the number line. This encompasses integers, fractions, and irrational numbers. Basically, any number that doesn't involve imaginary parts is a real number.Let's look at real numbers in terms of their relevance in function domains:

• Integers: Whole numbers like -2, -1, 0, 1, 2, \(...\).
• Fractions and Decimals: Numbers like 1/2, 0.75, -3.45.
• Irrational Numbers: Non-repeating, non-terminating decimals like \( \pi \) and \( \sqrt{2} \).

In our case of function subtraction and determining the domain for functions \( f(x) \) and \( g(x) \), all real numbers are valid inputs. This is because there are no mathematical operations, like divisions by zero or taking square roots of negatives, that would cause real number exclusions. Therefore, real numbers ensure broad applicability and flexibility in mathematical functions, especially for determining domains of quadratic functions like \( f-g \).