Problem 20 - 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. $$(f-g)(-2)$$

Step-by-Step Solution

Answer

The value of (f-g)(-2) is 7

Step 1: Substitution into functions

Substitute -2 into f: \(f(-2)=(-2)^{2}-1 = 4 - 1 = 3\). Then substitute -2 into g: \(g(-2)= -2-2 = -4\)

Step 2: Subtraction of function values

Now subtract the value obtained from function g from the value of function f at -2: \(f(-2)-g(-2) = 3 - (-4) = 3 + 4 = 7\)

Step 3: Final answer

The result of (f-g)(-2) after evaluating each function for x=-2 and performing subtraction is 7

Key Concepts

Function Subtraction

When it comes to evaluating the subtraction of two functions, we're essentially looking at the difference in their outputs for any given input. Imagine this as taking the outcome from one process and removing the impact of the other. In our exercise, we have two functions:

• Function: \( f(x) = x^2 - 1 \)
• Function: \( g(x) = x - 2 \)

To perform function subtraction, we want to compute \((f - g)(x)\) which means subtracting the value of \( g(x) \) from \( f(x) \). If we evaluate these at \( x = -2 \), we first determine \( f(-2) \) and \( g(-2) \),then take the difference:

• \( f(-2) = 3 \)
• \( g(-2) = -4 \)
• \((f - g)(-2) = 3 - (-4) = 7 \)

Through this subtraction, we find how much more or less the output of \( f \) is compared to \( g \) for that specific input.

Substitution Method

The substitution method is essential when evaluating functions at specific points. It involves replacing the variable in a function's expression with a particular value. In our example, to find the contribution of each function to the final subtraction result:1. Begin with substituting into \( f(x) \). For input \( x = -2 \), replace \( x \) in \( f(x) = x^2 - 1 \) with \(-2\).
- Calculate \( (-2)^2 - 1 = 4 - 1 = 3 \)2. Move to substituting into \( g(x) \) with the same \( x = -2 \).
- Calculate \( -2 - 2 = -4 \)By methodically applying values into these functions, substitution helps in constructing the numerical foundation needed for further function operations like subtraction.It's a straightforward method, simplifying the task of evaluating complex functions step-by-step.

Graphing Utility Verification

Using graphing utilities can be an excellent method to validate algebraic results, especially when dealing with function operations like subtraction. Let's break down how these tools help verify our mathematical findings:- **Visual Confirmation**: By plotting both functions \( f(x) = x^2 - 1 \) and \( g(x) = x - 2 \) on a graph, you can clearly see the points where they intersect. You can also identify their values at \( x = -2 \) visually.- **Cross-Verification**: Once plotted, a graphing utility will show their corresponding outputs at specific input values. It confirms the computed results from step-by-step algebraic solutions.
For example, at \( x = -2 \), look at the graph's curve for \( f(x) \) and \( g(x) \) to verify:- The point on \( f(x) \) should be at a vertical position of 3.- The point on \( g(x) \) should be at a vertical position of -4. - Subtracting these values graphically should also lead to the visual equivalent of our algebraic result, \( 7 \).This method provides a form of assurance, sticking a visual pin in our numerical conclusions, which can greatly aid understanding and bolster confidence in our math skills!