Problem 28 - 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. $$(f+g)(t-3)$$

Step-by-Step Solution

Answer

The function \((f+g)(t-3)\) evaluated algebraically is \(t^{2} - 5t + 3\).

Step 1: Evaluate the function \(f\) at \(t-3\)

To start, we evaluate the function \(f\) at the point \(t-3\). This means to substitute \(t-3\) in place of \(x\) in the function \(f(x)=x^{2}-1\). This results in \(f(t-3) = (t-3)^{2}-1\). After simplifying, this gives \(f(t-3) = t^{2} - 6t + 9 - 1 = t^{2} - 6t + 8\).

Step 2: Evaluate the function \(g\) at \(t-3\)

Next, the function \(g\) is evaluated at the point \(t-3\). This means to substitute \(t-3\) in place of \(x\) in the function \(g(x)=x-2\). This results in \(g(t-3) = (t-3) - 2\). After simplifying, this gives \(g(t-3) = t - 5\).

Step 3: Add the functions

Now that both functions have been evaluated at the point \(t-3\), the two results should be added together to get the final result: \((f+g)(t-3) = f(t-3) + g(t-3) = t^{2} - 6t + 8 + t - 5 = t^{2} - 5t + 3\).

Key Concepts

Function Evaluation

Understanding how to evaluate functions is crucial when studying algebra. In essence, function evaluation involves substituting a specified value into the function in place of the variable. For instance, given the functions f(x)=x^2-1 and g(x)=x-2, evaluating them at a particular input, say (t-3), is performed by replacing every instance of x in the expressions with (t-3). It’s like taking a snapshot of the function’s output at a specific point.

After the substitution, it is important to simplify the resulting expressions to make them easier to work with, especially before proceeding to any further operations such as function addition, as seen in the exercise with (f+g)(t-3). Understanding this concept is vital for solving a wide range of algebra problems involving functions.

Polynomial Simplification

Polynomial simplification is a fundamental skill in algebra that involves reducing expressions to their simplest form. We achieve this by combining like terms and applying algebraic operations such as addition, subtraction, and exponentiation rules. Take the expression obtained from evaluating f(t-3) from our example, t^2 - 6t + 9 - 1. Simplification involves combining the constants 9 and -1 to get t^2 - 6t + 8. Simplifying the expression ensures clarity and often makes it easier to graph or further manipulate the function.

When adding the simplified forms of f(t-3) and g(t-3), i.e., t^2 - 6t + 8 and t - 5, it is critical to line up like terms and then combine them, resulting in the simplified t^2 - 5t + 3. This step reduces the possibility of errors in subsequent calculations or interpretations.

Graphing Utility Verification

After algebraic function evaluation and simplification, it’s a good practice to use a graphing utility for verification. A graphing utility allows one to visualize functions, which serves as a powerful check against algebraic errors. By graphing the original functions f(x) and g(x), and their sum (f+g)(x), you can observe intersections and shape to understand their behavior.

In today's digital world, graphing utilities have become readily accessible and can provide immediate feedback. For educational purposes, they are invaluable in reinforcing algebraic concepts and fostering a deeper intuition for the graphical representation of functions. When used effectively, such tools complement and enhance understanding of the algebra involved by offering a visual perspective on the relationships between functions.