Notation

Example:

• This is the domain we have chosen.

• A sketch of this graph would look like this: (Note: This sketch shows the line starting from $x = 3$ and increasing.)

• However, we have only allowed the function f to be defined for $x > 3$.

• $f(3) = 2(3) + 3 = 9$ (Note: Point at $(3, 9)$ is marked, illustrating that the function starts from this point.)

• Notice, to illustrate the graph continues forever, the line has just vanished off the grid.

Example of an Incorrect Domain for a Real Function:

• This cannot be correct as the function is undefined for $x \leq -3$.

Example 1:

• Sketch: $y = x^2 + 2x + 7$ for $x \in \mathbb{R}$ and state its range.

• The sketch shows a parabola opening upwards with its vertex at $(-1, 6)$. This helps in determining the range.

• Calculator Steps:

• Input the quadratic formula.
• The discriminant shows no real roots, indicating the vertex is the minimum point.
• Minimum point calculation gives: $x = -1$ and $y = 6$.
• Thus, the range is $\xcancel {y \geq 6}$ or $f(x) \geq 6$.

Example 2:

• Find the range of $g(x) = e^x, x \geq 2$.

• Sketch shows the exponential function $y = e^x$ starting from $x = 2$.

• The blue highlight indicates the part of the function of interest.

• $g(2) = e^2$, indicating that $g(x) \geq e^2$ for $x \geq 2$.

• The range is the set of all $y$ values such that $y \geq e^2$.

The function $f$ is defined by $f(x) = 2 - \sqrt{x}$ for $x \geq 0$. The graph of $y = f(x)$ is shown above.

1. State the range of $f$.

• At $x = 0$:

Thus, $f(x) \leq 2$. The range of f is $f(x) \leq 2$.

2. Find the value of $ff(4)$.

• First, calculate $f(4)$:

• Then, calculate $ff(4) = f(0)$:

Example:

• Domain: $\{x : x \in \mathbb{R} \}$
• Range: $\{f(x) : f(x) \geq 3 \}$

From the above, we can construct the statement:

(beyond the space)

$f$ maps real numbers to real numbers greater than or equal to $3$.

Taking some specific values of mapping:

Domain: $\{-1, 0, 1, 2\}$

Range: $\{3, 4, 5, 7\}$

For $f$:

• $f(-1) = 4$
• $f(1) = 4$
• $f(2) = 7$ Notice that for each element of the range, there are two elements (i.e., many) that map to one element of the range.

For example, $f(-1) = f(1) = 4$.

Thus, the function is MANY-TO-ONE or MANY-ONE.

Example:

• Domain: $\{x : x \in \mathbb{R}\}$
• Range: $\{f(x) : f(x) \geq 3\}$

Since the line intersects the curve twice, we can see graphically that one y value is obtained from many $x$ values.

Example of one-to-many:

One $x$-value, e.g., $x = 0$, gives many $y$-values (i.e., $0, 6$).

Draw a vertical line to test for this. If the line cuts in more than one place at any point, the function is ONE-TO-MANY.