Question

determine if each represents a function. if it is not a function, explain why.

1. diagram of two - circles with arrows
2. {(-2,9),(2,0),(3,6),(4, - 1),(5,9)}
3. coordinate - plane with points
4. {(-3,2),(2,2),(1,2),(3,1),(0,3)}
5. coordinate - plane with a curve
6. table with x and y values

evaluate each expression with the given values:

1. (n^{2}-3n + 8), if (n = 4)
2. (\frac{2-4y}{z}), if (x = 4,y=-3)
3. (f(x)=4|x - 2|), find (f(-3))
4. find (f(-3)) for (y = f(x)) coordinate - plane with a curve

Explanation:

Step1: Determine if each represents a function

• For 1: In a function, each input (x - value) must have exactly one output (y - value). By vertical - line test (for graphs) or checking the mapping (for sets of ordered pairs), we analyze each case.
• For 2: The set of ordered pairs \(\{( - 2,9),(2,0),(3,6),(4, - 1),(5,9)\}\) is a function since each x - value has a unique y - value.
• For 3: The graph passes the vertical - line test, so it is a function.
• For 4: The set \(\{( - 3,2),( - 2,2),( - 1,2),(0,1),(0,3)\}\) is not a function because \(x = 0\) has two different y - values (\(y=1\) and \(y = 3\)).
• For 5: The graph fails the vertical - line test, so it is not a function.
• For 6: The table represents a function as each x - value has a single y - value.

Step2: Evaluate expressions

For 7:

Substitute \(n = 4\) into \(n^{2}-3n + 8\).
\[

$$\begin{align*} n^{2}-3n + 8&=(4)^{2}-3\times4 + 8\\ &=16-12 + 8\\ &=4 + 8\\ &=12 \end{align*}$$

\]

For 8:

Substitute \(x = 4\) and \(y=-3\) into \(\frac{2-4y}{2}\).
\[

$$\begin{align*} \frac{2-4y}{2}&=\frac{2-4\times(-3)}{2}\\ &=\frac{2 + 12}{2}\\ &=\frac{14}{2}\\ &=7 \end{align*}$$

\]

For 9:

Given \(f(x)=4|x - 2|\), substitute \(x=-3\).
\[

$$\begin{align*} f(-3)&=4|-3 - 2|\\ &=4|-5|\\ &=4\times5\\ &=20 \end{align*}$$

\]

For 10:

From the graph of \(y = f(x)\), when \(x=-3\), find the corresponding y - value. By looking at the graph, \(f(-3)=-7\).

Answer:

1. Analyze one - to - one mapping or vertical - line test for functions.
2. Function.
3. Function.
4. Not a function.
5. Not a function.
6. Function.
7. 12
8. 7
9. 20
10. - 7