Problem 37 - Precalculus Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. \(f(x)=\frac{x}{|x|}\) a. \(f(6)\) b. (-6) c. \(f\left(r^{2}\right)\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x}{|x|}\) evaluates to 1 when \(x = 6\), to -1 when \(x = -6\), and to 1 when \(x = r^2\).

Step 1: Evaluate the function \(f(x) = \frac{x}{|x|}\) for \(f(6)\)

For \(x = 6\), when we substitute 6 into \(x\), the function becomes \(f(6) = \frac{6}{|6|}\). Since \(|6|\) equals 6, this simplifies to \(f(6) = 1\).

Step 2: Evaluate the function for \(f(-6)\).

Substitute -6 into \(x\), we get \(f(-6) = \frac{-6}{|-6|}\). After simplifying, since \(|-6|\) equals 6, we have \(f(-6)= \frac{-6}{6}\) which equals -1.

Step 3: Evaluate the function for \(f(r^2)\)

Substituting \(r^2\) into \(x\), the function will be \(f(r^2) = \frac{r^{2}} {|r^{2}|}\). Any square of a real number is non-negative, so \(|r^{2}| = r^{2}\). Thus, we'll end up with \(f(r^2) = 1\).

Key Concepts

Absolute Value

When working with absolute values, it's essential to understand that the absolute value of a number is the distance of that number from zero on the number line, regardless of the direction. In mathematical terms, the absolute value function, denoted by two vertical bars |x|, transforms x into its positive counterpart if it is negative, and keeps it positive if it is already so. Essentially, it is a measure of magnitude without considering direction.

For instance, when evaluating an expression like |6|, we're simply looking at how far 6 is from zero. Since 6 is already positive, |6| equals 6. Conversely, |-6| is also 6, as -6 is six units away from zero as well but in the opposite direction. This concept is crucial in our exercise, particularly when we are simplifying expressions that include absolute value, like in the case of the function evaluation \(f(x) = \frac{x}{|x|}\).

Function Simplification

Function simplification refers to the process of reducing a complex expression into its simplest form. This often involves combining like terms, factoring, expanding expressions, and canceling common factors. Simplifying mathematical expressions can make it easier to evaluate, compare and further manipulate mathematical expressions.

Take our textbook example, where the function simplifies quite neatly. When we apply the absolute value rule, the functions \(f(6)\) and \(f(-6)\) simplify to 1 and -1 respectively, because after the absolute value of x is found, x and the absolute value of x become the same (if x is positive) or exact opposites (if x is negative). As such, the fraction simplifies to either \(\frac{x}{x}\) or \(\frac{-x}{x}\), yielding the simplified results of 1 and -1.

Substitution Method

The substitution method is a foundational tool in algebra that allows you to evaluate functions by substituting the given values of the independent variable into the function. This method goes hand in hand with function simplification and is particularly helpful when dealing with functions and complicated expressions.

In practice, if we have a function like \(f(x)\) and we want to evaluate it at \(x = a\) for some value a, we substitute a for every instance of x in the function. For example, with our function \(f(x)=\frac{x}{|x|}\) when evaluating \(f(r^2)\), we replace x with \(r^2\). Since we know absolute value measures magnitude, we ignore whether \(r^2\) was negative before squaring (it can't be after) so \(f(r^2)\) simplifies nicely to 1, demonstrating the effectiveness of this method in function evaluation.