Functions: Further Domain and Range, Modulus Function, Sketching Graphs

1. Domain and Range

1.1 Domain

The domain of a function is the set of all possible input values (usually denoted by $x$) for which the function is defined. It's crucial to consider restrictions on the domain, such as those arising from square roots, logarithms, or rational expressions.

Examples:

• $f(x) = \sqrt{x-2}$: The domain is $x \ge 2$, or $[2, \infty)$. We must have $x-2 \ge 0$.
• $f(x) = \frac{1}{x-1}$: The domain is $x \neq 1$, or $(-\infty, 1) \cup (1, \infty)$. We cannot divide by zero.
• $f(x) = \log(x-3)$: The domain is $x > 3$, or $(3, \infty)$. We must have $x-3 > 0$.

1.2 Range

The range of a function is the set of all possible output values (usually denoted by $y$) that the function can produce. It's the set of all $f(x)$ values for all $x$ in the domain.

Determining the range often involves considering the possible values of the function and any restrictions imposed by the domain.

Examples:

• $f(x) = \sqrt{x-2}$: The range is $y \ge 0$, or $[0, \infty)$. Since the square root is always non-negative.
• $f(x) = \frac{1}{x-1}$: The range is $y \neq 0$, or $(-\infty, 0) \cup (0, \infty)$. The function approaches 0 as $x$ approaches 1, but never equals 0.
• $f(x) = \log(x-3)$: The range is all real numbers, or $(-\infty, \infty)$. The logarithm can take any real value.

2. Modulus Function

2.1 Definition

The modulus function, denoted by $|x|$, is defined as:

$$ |x| = \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0 \end{cases} $$

The modulus function always returns a non-negative value.

2.2 Graphical Representation

The graph of $y = |x|$ is a V-shaped graph with its vertex at the origin (0, 0).

2.3 Modulus of a Function

The modulus of a function $f(x)$ is defined as $|f(x)|$. This means we take the absolute value of the function's output.

Example: If $f(x) = x-1$, then $|f(x)| = |x-1|$. The graph of $y = |x-1|$ is the graph of $y = x-1$ reflected across the x-axis.

3. Sketching Graphs

3.1 General Approach

1. Identify the domain and range: Determine any restrictions on the input and output values.
2. Find key points: Calculate the x and y intercepts (where the graph crosses the axes) and any turning points (local maxima or minima).
3. Consider asymptotes: Identify any vertical or horizontal asymptotes.
4. Plot the points: Plot the key points and asymptotes on a graph paper.
5. Draw the curve: Draw a smooth curve through the plotted points, taking into account the function's behavior.

3.2 Examples

3.2.1 $f(x) = \frac{1}{x-2}$

• Domain: $x \neq 2$, or $(-\infty, 2) \cup (2, \infty)$
• Asymptote: Vertical asymptote at $x = 2$
• x-intercept: None
• y-intercept: $f(0) = -1$, so the y-intercept is (0, -1)
• Turning points: None

3.2.2 $f(x) = \sqrt{x-1}$

• Domain: $x \ge 1$, or $[1, \infty)$
• x-intercept: $f(1) = 0$, so the x-intercept is (1, 0)
• y-intercept: $f(0)$ is undefined.
• Turning point: At x = 1, the function changes from undefined to 0.

3.2.3 $f(x) = |x-1|$

• Domain: All real numbers, or $(-\infty, \infty)$
• x-intercept: $f(1) = 0$, so the x-intercept is (1, 0)
• y-intercept: $f(0) = |0-1| = 1$, so the y-intercept is (0, 1)
• Turning point: At x = 1, the function changes direction.