2. Definition of a Function

Definition:

A function f: ℝ -→ ℝ associates to each number x in its domain exactly one number f(x).

---

Example 1:

H: ℝ -→ ℝ associates to each number t > 0 the height of a falling object t seconds after it was dropped.

Example 2:

P: ℝ -→ ℝ associates to each number T ≥ 0 the pressure of an ideal gas at temperature T Kelvins.

---

Falling Object Formula:

$$H(t)=h-\frac{1}{2}g{t}^2$$

So, for $h=200$ m and $g=10$ $m/s^2$

$$H(5)=200-\frac{1}{2}{\cdot }_{}10{\cdot }_{}{5}^2=75$$ $$H(3)=200-\frac{1}{2}{\cdot }_{}10{\cdot }_{}{3}^2=155$$

Ideal Gas Formula:

$$P(T)=\frac{nRT}{V}$$

---

Definition:

The graph of a function f: ℝ -→ ℝ is the curve going through all the points (x, f(x)) for x in the domain of f.

Graph Example:

$$FallingObject:H(t)=200-\frac{1}{2}{\cdot }_{}10{\cdot }_{}{t}^2$$

$$IdealGas:P(T)=45{\cdot }_{}8\cdot T$$

To find values from the graph, you can use a reading off way. For falling object, when t is equal to 5, then the output which is the height will be 75.

---

To figure out if a graph is a function or not, we use the Vertical Line test.

Theorem

A curve in the plane is the graph of a function over a certain domain if every vertical line at x-values in that domain intersects the curve exactly once.

Example 1:

Using the vertical line test, there is only 1 point on each vertical line, therefore making this a function.

Caution

A circle is not a function as it does not pass the Vertical Line test.