Graphs of Functions

A function is a relation in which each object in the domain has only one image. Two objects may share an image (many-to-one), so "each image has one object" is not required. Objects with more than one image make the relation not a function.

Each object has only one image, but two objects share the same image (−2 and 2 both go to 4; −1 and 1 both go to 1). That is a many-to-one relation, which is a function.

In {(2, 1), (2, 3), (4, 5)} the object 2 has two images, 1 and 3, so it is a one-to-many relation. In the other sets every first number appears only once, so each is a function (the first and last are many-to-one).

f(1) = 5, f(2) = 7 and f(4) = 2(4) + 3 = 11, so the range is {5, 7, 11}. {2, 4, 8} is the domain doubled (forgetting + 3), and {5, 7, 9} comes from using 3 for the last object instead of 4.

y = 2(−2)² − 5 = 2(4) − 5 = 3. −13 comes from treating (−2)² as −4. 11 comes from squaring 2 × (−2). −9 comes from not squaring: 2(−2) − 5.

y = 6x is a reciprocal function, and its graph is a hyperbola with two branches. y = 2x + 1 gives a straight line, y = x² − 4 a parabola and y = x³ + 2 a cubic curve.

The graph is a U-shaped parabola, so the function is quadratic. At x = 0, y = −4, and at x = 2, y = 4 − 4 = 0, which match y = x² − 4. 4 − x² would open downwards, and x² + 4 has its lowest point at (0, 4).

Draw a line across from y = 6 to the graph, then down to the x-axis: x = 4. Check: 2(4) − 2 = 6. 10 is the value of y when x = 6, and 2 is the value of y when x = 2.

y = 0 on the x-axis. The curve crosses the x-axis at x = 1 and x = 3. Check: 1 − 4 + 3 = 0 and 9 − 12 + 3 = 0. x = 2 is where the lowest point is, where y = −1.

At t = 4 the line is at C = 14, so he pays RM14. The line does not start at 0: there is a RM2 charge to enter, so 3 × 4 = 12 is not enough. RM11 is the charge for 3 hours.

The lines cross at (8, 4): both candles are 4 cm tall after 8 hours. Before that candle A is taller. Between 8 and 12 hours the line for B is above the line for A, so B is taller (at 10 hours A is 0 cm and B is 2 cm). After 12 hours both have burnt out.

The curve is at h = 5 when t = 1 and when t = 5 (check: 6 − 1 = 5 and 30 − 25 = 5). The ball is above 5 m from 1 s to 5 s, which is 5 − 1 = 4 seconds. 5 is only the second time the ball is at 5 m; 6 is the whole flight.