Question

Determine whether each relation represents a function.For each function, State the domain and the range.

(a) $\left\{\right(2,5),(4,6),(6,7),(8,8\left)\right\}$

(b) $\left\{\right(1,3),(4,-2),(-3,5),(1,7\left)\right\}$

(c)

(d)

Short answer

Part a. The relation is a function and its domain is given by the set $\left\{2,4,6,8\right\}$and the range is given by the set $\left\{5,6,7,8\right\}$.

Part b. The relation is not a function.

Part c. The relation is not a function.

Part d. The relation is a function whose domain is the set of all real numbers and its range is given by the set $[2,\infty )$.

Step-by-step solution

Part (a) Step 1. Check the ordered pairs

For the first relation, $\left\{\right(2,5),(4,6),(6,7),(8,8\left)\right\}$there are no ordered pairs with the same first element and different second elements.

So the given relation is a function.

The domain is the set of first elements of ordered pairs and it is given as $\left\{2,4,6,8\right\}$.

The range is given by the set of second elements of ordered pairs and it is given as $\left\{5,6,7,8\right\}$.

Part (b) Step 1. Check the ordered pairs 

The relation$\left\{(1,3),(4,-2),(-3,5),(1,7)\right\}$is not a function because there are two ordered pairs, $(1,3)$and $(1,7)$, that have the same first element and different second elements.

Part (c) Step 1. Perform vertical line test

For the given graph, we draw a vertical line as

So it can be seen that a vertical line intersects the graph at more than one point. So the graph is not the graph of a function.

Part (d) Step 1. Perform vertical line test

For the given graph, we draw a vertical line as

It can be seen that each vertical line intersects the graph at atmost one point.

So the given graph is the graph of a function.