Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f-g)(0)$$

Step-by-Step Solution

Answer

The value of \((f-g)(0)\) is 1.

Step 1: Define the Functions

We begin by defining the functions provided in the problem. For this problem, we have two functions \(f(x)=x^{2}-1\) and \(g(x)=x-2\).

Step 2: Create (f-g)(x)

We are looking for \((f-g)(0)\), so we need to create the function \(f-g\). To subtract one function from another, subtract the expressions that represent the two functions: \(f(x)-g(x) = x^{2}-1-(x-2) = x^{2}-1-x+2 = x^{2}-x+1\). So, \((f-g)(x)=x^{2}-x+1\).

Step 3: Substitute x=0 into (f-g)(x)

Now, we want to find \((f-g)(0)\), so we substitute \(x=0\) into our new function \((f-g)(x)\). This yields: \((f-g)(0)=0^{2}-0+1=1\).

Step 4: Confirm with Graphing Utility (Optional)

If available and allowed, one can use a graphing utility to check this result. This step involves graphing the function \(f(x)-g(x)\) (which simplified to \(x^{2}-x+1\)) and observing the y-value when \(x=0\). The function graph should intersect the y-axis at \(y=1\), confirming the answer.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition or subtraction. When you see an expression like \(x^2 - 1\), it represents a mathematical phrase involving a variable "\(x\)". These expressions can vary in complexity, from simple terms with numbers and variables to more complicated forms with multiple terms.

For example, in the given functions, \(f(x) = x^2 - 1\) and \(g(x) = x - 2\), each function is its own unique algebraic expression. To combine these functions, you can perform operations like addition, subtraction, multiplication, or division.

This process helps in forming new expressions, allowing us to explore different mathematical properties and relationships.

Exploring Polynomial Functions

Polynomial functions are a type of algebraic expression characterized by terms consisting of variables raised to whole number powers. An example of a polynomial function is \(f(x) = x^2 - 1\). This is because it includes a term \(x^2\), which is a variable "\(x\)" raised to a power of 2.

Key features of polynomial functions include:

• Degree: The highest power of the variable, which determines the function's behavior.
• Coefficients: Numbers multiplied by the variable terms, like the "1" in \(x^2\).
• Constants: Standalone numbers, such as "\(-1\)" in \(f(x)\).

By understanding these components, you can manipulate and transform polynomial functions to create new expressions, like finding \((f-g)(x) = x^2 - x + 1\) in our example.

Performing Function Evaluation

Function evaluation is the process of calculating the output of a function for specific input values. In our exercise, we evaluated the function \((f-g)(x)\) at \(x = 0\).

Here's how you can do this:

• First, form the expression or function you wish to evaluate, like \((f-g)(x)=x^2-x+1\).
• Substitute the given input value (here, \(x=0\)) into the function.
• Simplify the expression to determine the output, which in this case is \((f-g)(0) = 1\).

This process is essential in understanding how functions behave under different conditions and helps verify solutions, such as using a graphing utility if possible.