Problem 31 - 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. $$(f / g)(t-4)$$

Step-by-Step Solution

Answer

The evaluation of \((f / g)(t-4)\) is \((t^{2}-8t+15) / (t-6)\).

Step 1: Evaluate function f at t-4

We replace \(x\) in \(f(x)=x^{2}-1\) with \(t-4\) to get \(f(t-4)=(t-4)^{2}-1=t^{2}-8t+16-1=t^{2}-8t+15.\n

Step 2: Evaluate function g at t-4

We replace \(x\) in \(g(x)=x-2\) with \(t-4\) to get \(g(t-4)=(t-4)-2=t-6.\n

Step 3: Compute (f / g)(t-4)

We divide the obtained f(t-4) by g(t-4) to compute (f / g)(t-4). Hence, \((f / g)(t-4)=(t^{2}-8t+15)/(t-6)\n.

Step 4: Verification using a graphing utility

Plot both functions \(f(t-4)\) and \(g(t-4)\) using a graphing utility, as well as the division of both functions. Make sure all functions overlap at all values of \(t\), excluding \(t = 6\), as \(g(t-4)\) is not defined at this point.\n

Key Concepts

Algebraic Function Evaluation

Understanding algebraic function evaluation is essential when working with functions in mathematics. It involves finding the output of a function for a given input. To illustrate with our example, evaluating the function f for an input of t-4 requires us to substitute t-4 into the function's formula. Here's how it can be broken down:

When we substitute t-4 into f(x)=x^2-1, we replace every instance of x with t-4. This substitution gives us f(t-4)=(t-4)^2-1. Expanding (t-4)^2 results in t^2 - 8t + 16, and subtracting 1 from that we get f(t-4) = t^2 - 8t + 15.

Similarly, for function g, substituting t-4 into g(x)=x-2, yields g(t-4)=(t-4)-2 = t-6. In both cases, the result is a new expression that displays how the function behaves for the input t-4. This process is crucial for understanding how functions change with different inputs and is the first step towards mastering function composition.

Rational Expressions

Rational expressions are fractions in which the numerator and the denominator are polynomials. In the provided exercise, after evaluating the functions f and g at t-4, we arrive at a rational expression: \[\frac{t^{2}-8t+15}{t-6}\].

The numerator, t^2 - 8t + 15, and the denominator, t-6, are both polynomials, creating the rational expression or function. It's important to simplify such expressions and identify any restrictions on the variable t. For this example, we must note that the expression is undefined when t-6=0, which occurs at t=6. Understanding undefined conditions is crucial to avoid errors in calculations and interpretations of rational expressions.

Simplifying Rational Expressions

Although not applicable in this specific exercise, simplifying rational expressions generally involve factoring the numerator and denominator and then canceling common factors. However, since the expression \(t^2 - 8t + 15\) cannot be factored further to cancel terms with \(t-6\), this rational function is already in its simplest form. Recognizing when expressions are in simplest form helps streamline solving equations and improves algebraic manipulation skills.

Graphing Utility Verification

Graphing utilities are powerful tools that help verify algebraic work and provide visual insights into functions' behaviors. For our specific function (f / g)(t-4), a graphing utility helps confirm the accuracy of our algebraic evaluation by visually checking the graphs of f(t-4) and g(t-4), as well as their quotient.

By plotting f(t-4) and g(t-4) on the graphing utility, a well-defined curve and a straight line, respectively, should appear. To check our rational function (f / g)(t-4), we graph the corresponding expression and look for points of discontinuity or undefined values. In this case, we expect the graph to break or have a hole at t=6 since g(t-4) is undefined at that point.

Advantages of Graphing Utilities

Graphing utilities can also depict asymptotic behaviors, intercepts, and intervals of increase or decrease—important features that might not be immediately apparent through algebraic manipulation alone. Furthermore, these tools facilitate the understanding of complicated expressions and provide immediate feedback on the correctness of the algebraic process. Utilizing them reinforces comprehension and is a way of double-checking one's work before arriving at a final solution.