Question

1. $f(x) = x^2 - 2x - 4$; find $f(3)$

(a) determine if the relation is a function.
(b) explain.
30)
graph with x and y axes, grid, and a piecewise linear graph: one line from (-8, -6) to (0, 2), another from (0, 2) to (-8, 8)

Explanation:

Problem 28: Find \( f(3) \) for \( f(x) = x^2 - 2x - 4 \)

Step 1: Substitute \( x = 3 \) into the function

To find \( f(3) \), we replace every \( x \) in the function \( f(x) = x^2 - 2x - 4 \) with 3. So we get \( f(3) = (3)^2 - 2(3) - 4 \).

Step 2: Simplify the expression

First, calculate the powers and multiplications: \( (3)^2 = 9 \), \( 2(3) = 6 \). Then substitute these back into the expression: \( f(3) = 9 - 6 - 4 \). Now perform the subtractions: \( 9 - 6 = 3 \), and then \( 3 - 4 = -1 \).

To determine if a relation (represented by a graph) is a function, we use the Vertical Line Test. The Vertical Line Test states that if any vertical line drawn through the graph intersects the graph at more than one point, then the relation is not a function.

Looking at the given graph, if we draw a vertical line (for example, the vertical line \( x = 0 \) or any vertical line that would pass through the "corner" or other parts of the graph), we can see that there are vertical lines that would intersect the graph at more than one point. Specifically, the graph has a "corner" or a point where two lines meet, and for some \( x \)-values (around the \( x \)-value of the corner), a vertical line would cross the graph twice.

Answer:

\( f(3) = -1 \)

Problem 30 (a) and (b): Determine if the relation is a function and explain