5.6.8 - Quick Check: Formalizing Relations and Functions

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The step-by-step solution for the questions above are as follows:

1. What is the domain of the function below? {(0, 2), (3, 1), (5, 2), (8, 4)}?
- The domain of a function is the set of all possible x-values.
- In this case, the x-values are 0, 3, 5, and 8.
- Therefore, the domain is {0, 3, 5, 8}.
- Correct answer: {0, 3, 5, 8}.

2. Is the following relation a function? {(0.3, 0.6), (0.4, 0.8), (0.3, 0.7), (0.5, 0.5)}
- To determine if a relation is a function, we need to check if each x-value corresponds to only one y-value.
- In this case, we have (0.3, 0.6), (0.4, 0.8), (0.3, 0.7), and (0.5, 0.5).
- The x-value 0.3 is paired with both 0.6 and 0.7, which means it does not satisfy the one-to-one correspondence required for a function.
- Therefore, the relation is not a function.
- Correct answer: no.

3. Use the vertical line test to determine whether the relation graphed below is a function.
- The vertical line test involves drawing vertical lines on the graph and checking if any of them intersect the graph more than once.
- If a vertical line intersects the graph more than once, it means that there are multiple y-values for a single x-value, which violates the definition of a function.
- In this case, there are no vertical lines that intersect the graph more than once.
- Therefore, the relation graphed is a function.
- Correct answer: yes.

4. Sound travels about 343 meters per second. The function d(t) = 343t gives the distance d(t) in meters that sound travels in t seconds. How far does sound travel in 8 seconds?
- To find the distance traveled in 8 seconds, we need to evaluate the function d(t) = 343t at t = 8.
- Substitute t = 8 into the function: d(8) = 343 * 8 = 2744.
- Therefore, sound travels 2744 meters in 8 seconds.
- Correct answer: 2,744 meters.

5. What is the value of the range of the function f(x) = x² + 2 for the domain value 1/4?
- The range of a function is the set of all possible y-values.
- To find the value of the range for a specific domain value, we need to substitute that value into the function and simplify.
- In this case, we are given the domain value x = 1/4.
- Substitute x = 1/4 into the function: f(1/4) = (1/4)² + 2 = 1/16 + 2 = 2 1/16.
- Therefore, the value of the range for the domain value 1/4 is y = 2 1/16.
- Correct answer: y = 2 1/16.

To find the domain of a function, you need to determine all the possible input values (x-values) that the function can take. In this case, the domain is the set of x-values in the given function.

For the function {(0, 2), (3, 1), (5, 2), (8, 4)}, you can see that the x-values are 0, 3, 5, and 8. Therefore, the domain of this function is {0, 3, 5, 8}.

To determine whether a relation is a function, you need to check if each input (x-value) is associated with only one output (y-value). If any x-value has multiple y-values, then the relation is not a function.

In the case of the relation {(0.3, 0.6), (0.4, 0.8), (0.3, 0.7), (0.5, 0.5)}, you can see that the x-value 0.3 is associated with both 0.6 and 0.7. Hence, the relation is not a function.

To determine whether a graph represents a function, you can use the vertical line test. Draw vertical lines on the graph, and if any vertical line intersects the graph at more than one point, then the graph does not represent a function. If all vertical lines intersect the graph at most at one point, then the graph represents a function.

In this case, the graph passes the vertical line test because no vertical line intersects the graph at more than one point. Thus, the graph represents a function.

For the distance function d(t) = 343t, you can plug in the value of t (8 seconds) into the equation to find the distance traveled.

d(8) = 343 * 8 = 2,744 meters. Therefore, sound travels 2,744 meters in 8 seconds.

To find the range of a function for a specific x-value, you need to substitute the x-value into the function and solve for the corresponding y-value.

For the function f(x) = x² + 2, the domain value is given as 1/4.

Substituting x = 1/4 into the function:
f(1/4) = (1/4)² + 2
f(1/4) = 1/16 + 2
f(1/4) = 33/16 = 2 1/16

Therefore, the value of the range for the given domain value 1/4 is y = 2 1/16.