Problem 5 - The Calculus with Analytic Geometry

Question

Given $$ f(x)= \begin{cases}\frac{|x|}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases} $$ find: (a) \(f(1) ;\) (b) \(f(-1) ;\) (c) \(f(4) ;\) (d) \(f(-4) ;\left(\right.\) e) \(f(-x)\); (f) \(f(x+1) ;\) (g) \(f\left(x^{2}\right) ;\left(\right.\) h) \(f\left(-x^{2}\right)\).

Step-by-Step Solution

Answer

a) 1; b) -1; c) 1; d) -1; e) -f(x); f) \(\frac{|x+1|}{x+1}\); g) 1; h) -1.

Step 1 - Evaluate \(f(1)\)

Since \(1eq 0\), use the first part of the piecewise function: \(f(x) = \frac{|x|}{x}\). When \(x = 1\), \(|1| = 1\), so \(f(1) = \frac{1}{1} = 1\).

Step 2 - Evaluate \(f(-1)\)

Since \(-1eq 0\), use the first part of the piecewise function: \(f(x) = \frac{|x|}{x}\). When \(x = -1\), \(|-1| = 1\), so \(f(-1) = \frac{1}{-1} = -1\).

Step 3 - Evaluate \(f(4)\)

Since \(4eq 0\), use the first part of the piecewise function: \(f(x) = \frac{|x|}{x}\). When \(x = 4\), \(|4| = 4\), so \(f(4) = \frac{4}{4} = 1\).

Step 4 - Evaluate \(f(-4)\)

Since \(-4eq 0\), use the first part of the piecewise function: \(f(x) = \frac{|x|}{x}\). When \(x = -4\), \(|-4| = 4\), so \(f(-4) = \frac{4}{-4} = -1\).

Step 5 - Evaluate \(f(-x)\)

Use the first part of the piecewise function: \(f(x) = \frac{|x|}{x}\). When \(xeq 0\), \(f(-x) = \frac{|-x|}{-x} = \frac{|x|}{-x} = -\frac{|x|}{x}\). This implies: for \(x > 0\), \(f(-x) = -1\); for \(x < 0\), \(f(-x) = 1\).

Step 6 - Evaluate \(f(x+1)\)

Consider two cases based on the value of \(x + 1\). If \(x + 1 eq 0\), then \(f(x+1) = \frac{|x+1|}{x+1}\); otherwise \(f(x+1) = 1\) if \(x + 1 = 0\), which means \(x = -1\).

Step 7 - Evaluate \(f(x^2)\)

Since \(x^2 \geq 0\) and \(x^2 eq 0\) for all \(x eq 0\), \(f(x^2) = \frac{|x^2|}{x^2} = 1\). When \(x = 0\), \(f(0) = 1\). Therefore, \(f(x^2) = 1\).

Step 8 - Evaluate \(f(-x^2)\)

Since \(-x^2 \leq 0\) and \(-x^2 eq 0\) for all \(x eq 0\), \(f(-x^2) = \frac{|-x^2|}{-x^2} = \frac{x^2}{-x^2} = -1\). When \(x = 0\), \(f(0) = 1\). Therefore, \(f(-x^2) = -1\).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It's written as \(|x|\) and represents the distance of a number from zero on the number line, without considering direction. So, \(|x|\) always returns a non-negative value. For example, both \(|-4|\) and \(|4|\) are equal to 4.

The absolute value function is particularly useful in piecewise functions as it can simplify expressions and aid in understanding behaviors of functions across different intervals.

In our given problem, the expression \(\frac{|x|}{x}\) is part of a piecewise function that behaves differently based on whether \(x\) is positive or negative. Understanding how \(|x|\) removes the sign is crucial to solving the function evaluations provided in the problem.

Function Evaluation

Function evaluation involves finding the output of a function given a specific input. It is essential in understanding how functions behave. Let's break down the process using the piecewise function given in the problem:

• Identify which part of the piecewise function to use based on the input value. In our example, when \(x eq 0\), we use \(f(x) = \frac{|x|}{x}\).
• Substitute the input into the chosen part of the function and perform the calculations. For instance, when \(x = -4\), we first find \(|-4| = 4\) then calculate \(f(-4) = \frac{4}{-4} = -1\).

By following these steps, you can systematically evaluate any function, including more complex cases. For input transformations like \(f(x+1)\) or \(f(x^2)\), first compute the transformed input before applying the function.

Algebra

Algebra is the foundation of understanding and manipulating mathematical expressions and equations. In the context of this problem, several algebraic principles are at play:

• Simplifying expressions: Knowing that \(\frac{|x|}{x}\) simplifies based on the sign of \(x\).
• Handling piecewise functions: Recognizing which part of the function to use for different intervals or conditions of \(x\).
• Transformation of variables: For instance, evaluating \(f(x+1)\) involves shifting the input by one unit before using the piecewise function.

Mastering these algebraic techniques allows students to not only solve textbook problems but also gain deeper insights into how mathematical functions operate. For example, transforming inputs like \(x+1\) or interpreting complex behaviors in piecewise functions become more intuitive when understanding these algebraic concepts.