Concept: Function Evaluation and Properties of the Reciprocal Function

The function $f(x) = \frac{1}{x}$ assigns to each nonzero real number its multiplicative inverse. We evaluate it by direct substitution.

Step 1: Compute $f(x)$ for each given $x$-value:

• $f(-2) = \frac{1}{-2} = -0.5$
• $f(-1.5) = \frac{1}{-1.5} = -\frac{2}{3} \approx -0.667$
• $f(-1) = \frac{1}{-1} = -1$
• $f(-0.5) = \frac{1}{-0.5} = -2$
• $f(0.25) = \frac{1}{0.25} = 4$
• $f(0.5) = \frac{1}{0.5} = 2$
• $f(1) = \frac{1}{1} = 1$
• $f(1.5) = \frac{1}{1.5} = \frac{2}{3} \approx 0.667$
• $f(2) = \frac{1}{2} = 0.5$

Understanding the Reciprocal Function

The function $f(x) = \frac{1}{x}$ is one of the most fundamental non-linear functions in mathematics. It takes any non-zero number and returns its multiplicative inverse. The reason we exclude zero from the domain is simple: division by zero is undefined in the real number system.

Think about what this function does geometrically. For positive inputs, it returns positive outputs, but with an interesting twist: large inputs give small outputs and vice versa. When $x = 1$, we get $f(1) = 1$; when $x = 2$, we get $f(2) = \frac{1}{2}$, which is smaller. For negative inputs, the function behaves similarly but stays in the negative realm.

Completing the Table

To fill in the table, we substitute each $x$-value into the function $f(x) = \frac{1}{x}$.

1. For $x = -2$:

   $f(-2) = \frac{1}{-2} = -0.5$

2. For $x = -1.5$:

   $f(-1.5) = \frac{1}{-1.5} = \frac{1}{-\frac{3}{2}} = -\frac{2}{3} \approx -0.667$

3. For $x = -1$:

   $f(-1) = \frac{1}{-1} = -1$

4. For $x = -0.5$:

   $f(-0.5) = \frac{1}{-0.5} = \frac{1}{-\frac{1}{2}} = -2$

5. For $x = 0.25$:

   $f(0.25) = \frac{1}{0.25} = \frac{1}{\frac{1}{4}} = 4$

6. For $x = 0.5$:

   $f(0.5) = \frac{1}{0.5} = \frac{1}{\frac{1}{2}} = 2$

7. For $x = 1$:

   $f(1) = \frac{1}{1} = 1$

8. For $x = 1.5$:

   $f(1.5) = \frac{1}{1.5} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \approx 0.667$

9. For $x = 2$:

   $f(2) = \frac{1}{2} = 0.5$

Completed Table

Domain and Range …