Problem 26 - 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

So, the value of \((f / g)(0)\) is 0.5. This final result means that when function f(x) is divided by function g(x) at x equal to 0, the result is 0.5.

Step 1: Identifying Functions f and g

From the given exercise, we have two functions, that is, \(f(x) = x^{2} - 1\) and \(g(x) = x - 2\). Now, we need to substitute the given x value, which is 0, into each function for evaluation.

Step 2: Substituting x = 0 in both Functions

Substitute x=0 into the \(f(x)\) function gives \(f(0) = (0)^{2} - 1 = -1\). Similarly, when x=0 is substituted into the \(g(x)\) function it results in \(g(0) = 0 - 2 = -2\) .

Step 3: Dividing f(0) by g(0)

Finally, the result for \(f(0) / g(0) = -1 / -2 = 0.5\), this substitution results in division of two negative numbers, which results in a positive number.

Step 4: Verification Using a Graphing Utility

Although the exercise does not require to graphically verify the solution, if possible, this can be done by plotting the graphs of both \(f(x) = x^{2} - 1\) and \(g(x) = x - 2\) and checking the value of \(f(x) / g(x)\) at x = 0. It should match with our calculated value, that is, 0.5.

Key Concepts

Substituting in Functions

Substituting in functions is a fundamental skill necessary for evaluating expressions where you replace the variable with a specified value. This step is crucial for understanding how functions behave at particular points.

For example, let's say we have a function, such as in our exercise, where we are given f(x) = x^2 - 1. When asked to find the value of the function when x is 0, we substitute 0 for every instance of x. Thus, f(0) becomes 0^2 - 1, which simplifies to -1. This means that at x = 0, the output or value of the function f is -1. Understanding this concept helps students grasp the idea that functions are essentially rules that take an input and produce an output.

Dividing Functions

Dividing functions involves calculating the quotient of two functions. It requires the student to substitute values into each function and then perform division on the resulting outputs.

In the context of our problem, after substituting x = 0 into both functions, f and g, to get f(0) and g(0), the next step is to divide f(0) by g(0). Here, we calculate (f / g)(0) by dividing -1 (the value of f(0)) by -2 (the value of g(0)) to achieve a quotient of 0.5. Students should be made aware that dividing functions is permissible as long as the divisor, in this case, g(0), is not equal to zero, as division by zero is undefined in mathematics.

Graphing Utility Verification

Graphing utilities are powerful tools that can verify algebraic solutions by visual representation. Once a student solves an equation algebraically, like we computed (f / g)(0), they can use a graphing utility to plot the functions and visually confirm the computed value.

For our exercise, plotting f(x) = x^2 - 1 and g(x) = x - 2, and then finding the value of f(x) / g(x) when x = 0 should result in the same answer we obtained: 0.5. This not only reinforces the algebraic solution but also demonstrates the intersection of algebra and graphing techniques. It's also a helpful check to prevent errors, as any discrepancy between the graphical and the algebraic solutions would necessitate a reevaluation of the steps taken.