1. Algebra 1
Section 5.3

2. Graphs
t is the independent variable.
h is the dependent variable.

3. Graphs
D = {t | 0 ≤ t ≤ 3}
R = {h | 6 ≤ h ≤ 20}

4. Vertical Line Test
The vertical line test tells us that if a vertical line intersects the relation’s graph in more than one point, the relation is not a function.

5. Example 1a
D = {0, ±1, ±2, ±3}
R = {0, 0.5, 2, 4.5}
The relation is a function; no vertical line intersects the graph at more than one point.

6. Example 1b D = {0, ±1, ±2, ±3} R = {-2, -1, 0, 1, 2, 3, 4, 5}
The relation is not a function; a vertical line crossing the x-axis at -1 contains both (-1, -2) and (-1, 3).

7. Example 1c D = {x | x ≥ 0} R = all real numbers
The relation is not a function; any vertical line crossing the positive x-axis crosses the graph at more than one point.

8. Equations
An equation is often used to clearly define the rule that determines the values of the range associated with each member of the domain.

9. Example 2 A = {(x, y) | y = x + 2; x = 1, 2, 3, 4}
x y = x ordered pair 1 2 3 4 3 4 5 6
(1, 3)
(2, 4)
(3, 5)
(4, 6)

10. Example 2 A = {(x, y) | y = x + 2; x = 1, 2, 3, 4} y (1, 3) (2, 4)
(3, 5)
(4, 6) x

11. Example 2 A = {(x, y) | y = x + 2; x = 1, 2, 3, 4} y
This relation is a function.
No vertical line will intersect more than one point. x

12. What if...? A = {(x, y) | y = x + 2; x = 1, 2, 3, 4}
What if no domain had been stated for this function? x

13. What if...? A = {(x, y) | y = x + 2} y
Then the entire line would satisfy the equation. x

14. What if...? A = {(x, y) | y = x + 2} y
The coordinates of every point on the line satisfy the equation. x

15. Example 3
If x is just an integer from 1 to 5, then the graph consists of just five points.
If x can be any number from 1 to 5, then the graph would be a segment from (1, 3) to (5, 3).
In either case, it is a function.

16. Example 4 D = {(x, y) | y = ± x; x = 1, 4, 9} Domain: D = {1, 4, 9}
Range: R = {-3, -2, -1, 1, 2, 3}
This relation is not a function.
It fails the vertical line test.

17. Graphing Functions
Because functions are relations, they are graphed using the same methods.
If the domain is not restricted, then a smooth curve connecting the ordered pairs should be drawn.

18. Example 5
f(x) = -x2
(-2, -4)
(-1, -1)
(0, 0)
(1, -1)
(2, -4) y x

19. Example 5 The domain is all real numbers. The range is {y | y ≤ 0}. y

20. Homework: pp